I wanted to talk about this topic in response to every person under the sun who is complaining about counter magic. Before I go about saying anything, I want to let all you readers out there know that you have to acknowledge the existence of the blue spells that say "counter target spell". There is no avoiding it. Even if you are lucky, you would not be able to avoid them because they are so good at what they do.

Introduction: The card Counterspell has been around since the dawn of Magic and with it's birth came the ranting of players who disliked the fact that they could not play the spells they wanted. To be honest, I like playing Counterspells in any deck I can play them in.  Reason? Counter magic is the most efficient way of not losing. Not losing? How's that different from winning? When you are winning, you are closer to actually finishing the game. However, when you are not losing, you actually have more time to play your game. Counter magic cannot win games by themselves. They have a support role of protecting you and protecting your game plan. For this simple reason, great finishers have been born in the color of the sea because somebody in R&D probably figured out that Mahamoti Djinn as a win condition could have been improved.

During the recent years, our favorite counter magic has been the target of abuse. You don't get two mana "Counter target spell" anymore. You now have to pay three mana for that. Also, the enemies of blue have been getting powerful cheap creatures like Kird Ape, Mogg Fanatic and even Tarmogoyf. Heck, times have become so bad that Remove Soul is now actually great in standard. (Actually, Remove Soul is good because everyone is playing creatures and everyone is playing creatures because Planeswalkers are annoying if you can't use creatures to slap them away.) 

Blue has never lost its fighting edge, except maybe in the OLS season where big fat monsters stomped everywhere in the block format, because we have been given a few goodies to tide through tough times with. In fact, the other times when blue was good didn't really involve blue having counter magic. Cards like Thirst for Knowledge and Gifts Ungiven are such cards that thrived in the era of minimal counter magic, just to name a few. The more recent blocks have gained at least one great blue counter magic each (even if they might not be hard counters).

Examples: 
Ravnica block:

Remand looks really simple and cannot counter a spell for nuts even if you had 2 million copies of it in your deck yet it was given recognition as one of the best spells in the set for its super tempo gaining ability. 

Time Spiral Block:

Venser, Shaper Savant is a great "counter" against anything that is on the stack that cannot be recast on the same turn. It's as versatile as any blue card can get. 

Lorwyn Block: 

Cryptic Command is another ultra versatile card that is limited by your imagination and is the best counterspell in the world of faeries tricking goblins with mushrooms.

Collectively, throughout the years, blue mage has picked up an arsenal of great cards, even if they don't necessarily mean more than one every year. In Singleton, we learn to appreciate any addition to the card pool.

Away from all this talk about blue cards, let's get back to the main topic. How do we fight the annoying blue cards?

Anti-counter tech #1: Play More" Must Counter" Cards

A long time ago, counter decks meant a critical count of counter magic, like eighteen or more in a deck. Nowadays, a counter deck has evolved into good card drawing put together with some menacing counter magic. If you just keep casting spells that have to be countered, the opponent will have to let some through because the cards left in hand would be the "build up" cards.

What I am saying is that counter decks do not have the capacity to trade one for one (which is what most counterspells do) with your threats forever. To overcome this, they have to play cards that let them restock their hand or at least allow them to be able to trade more cards for their single card. If your every threat is so threatening, there would be no realistic way for the blue player to win because you will be able to force something through at some point in time when they run out of counter magic.

Anti-counter tech #1A: Unrivalled Card Drawing #1A is named so because these cards fall in the "must counter" category. As stated above, counter based decks trade cards at a 1 to 1 ratio most of the time. If you can find a way to out-draw your opponent (essentially draw more cards), you will be have more spells than the number of counter magic available in your opponents' hand. For such a reason, a turn two Dark Confidant is almost a game sealed against counter based decks that are not aggressive in their reaction (like having immediate spot removal before it becomes active). Phyrexian Arena or Jace Beleren fall into this same category where if resolved, they can just bring the game away from the blue mage who will not be able to keep up with the quantity of cards.

Once upon a time, in a land far far away, people were saying this card was bad.

Anti-counter tech #1B: Play Token Producers

By letting a token producer resolve, that one card suddenly becomes worth more. The threat from this type of card comes in two approaches. The first is the obvious fact that new tokens can be produced (which is usually at least once a turn). The second comes from the fact that there are creatures in play that threaten the life total. Once it becomes active, you cannot clean it up by just dealing with one aspect of it. You have to deal with both and not many cards can do both.

Bitterblossom from Morningtide is going to be brutal because of its casting cost and the lack of need to have any additional mana commitments to generate tokens.

Some token producing enchantments to take note of: Mobilization, Sacred Mesa and Squirrel Nest.
The most played token producer: Meloku the Clouded Mirror

Just to mention, token producers are quite decent against the aggressive mage as well. The only problem that might arise would be that token producers usually require a considerable amount of resources to get going so usually token producers are played when the control players have stabilized the board (at that point, any win condition should suffice), serving as a win condition that can hold the ground but would end up using more time, which is usually the reason why people tend to dislike using token producers.

Anti-counter tech #1C: Play Untargetable Cards

Counter based decks have traditionally worked like this: Counter some spells, bounce those that were "leaked" and counter them all over again, draw into bomby creature and win.

By playing untargetable cards, you are forcing them to stop your cards at the first level of defense because once they "leak", your guys will stay there for awhile unless they play non-targetted removals which tend to be "slow" and will allow you to get the more problematic guys into play.

Troll Ascetic and Nimble Mongoose are good cards that fall in this category.

Also, protection from blue guys can be considered "untargetable". Yavimaya Barbarian was played in the main deck of "Steroids", a RG deck similar to Fires without the signature Fires of Yavimaya card but was simply stuffed with RG goodness. The barbarian was a simply 2 drop against non-blue decks but a beating against blue decks when they could no longer Repulse it or block using creatures with the blue mana symbol in their casting costs.

Anti-counter tech #2: Play Persistent Cards Academy Ruins is my favorite card that falls into this category. If you keep putting this great card on the top of your library, you are forcing your opponent to counter the same card again and again and again until he eventually cannot do so. Your opponent would be forced to play Dissipate or Faerie Trickery in the main deck to have to deal with such cards. Three mana counterspells are not as efficient as they can get but happen to be the necessary evil nowadays. Another big headache for the counter mage would be Genesis. Genesis allows you to recur your problematic creatures again and again and again. For some reason, I had an opponent complaining about counters while having Genesis in the graveyard for almost half the game and he didn't bother to use it (my deck ran just a pitiful amount of counters btw). For those of you out there who have no idea how to fight counter mages, just recur creatures from the graveyard actively and cast the cards they have seen. Of course, this might not be the best plan 100% of the time but usually, if you keep doing that, he will have to let some through because he cannot counter everything under the sun, especially when you are almost drawing two cards a turn.

Speaking of persistence, dredge cards win hands down. Mr. Life from the Loam, I'm talking about you.

Anti-counter tech #3: Play Uncounterable Cards

Play uncounterable cards to fight counter magic? Ok... (That's not very new)

Uncounterable cards are powerful because they slip right through the counter barrier and it's going to be really difficult for the blue mage to plan against them because they would usually plan to actually counter some cards at certain points in time.

Just remember that the split second mechanic is as good as "card A is uncounterable", except when against Counterbalance. If you ever meet one, play a decoy spell before showing your business spell.

Sulfur Elemental always comes at EOT (end of turn) and connects for three against our island loving friends, Sudden Shock has sealed more games than any other card I've seen because there is almost nothing you can do about it except Willbender it away and Demonfire is the Fireball of choice to punch through the last points of damage. It must be very cool to say "no" to "no". 
(Why are the examples all red cards? Red benefits the most from uncounterability because it is a color that wants to force damage through. Other colors have better ways of fighting the blue mage which may sound incredibly odd.)

Sidetalk: The first "cannot be countered" card I've ever played was Kavu Chameleon. It's a very cool 5 mana 4/4 which at that time I thought was the best thing I could play (I started playing around then). It couldn't be countered or Terrored. Wow! Of course, no one plays it anymore but you definitely could if you're desperate enough.

Anti-counter tech #4: Play Cards with Low Casting Costs The most powerful characteristic of Counterspell is that it allows the caster to trade two blue mana and a card for any other card that may even cost 10,000,000 mana to cast. This therefore allows for a better usage of mana at any single point of the game. By playing cards like Kird Ape or Mogg Fanatic, you turn that condition around. They pay two mana and the spell for your one mana and the creature card. That seems totally wrong.  When you cast your one drops and your opponent has counter magic, they have to contemplate about whether or not you might have anything more threatening coming up next because most of the time they only have enough mana for a piece of counter magic. To be fair, the blue decks have evolved considerably and have tools to fight these strategies. Simply casting an Epochrasite, Engineered Explosives or Vedalken Shackles can thwart this plan quite abit. AEther Vial is the most brutal card that falls in this category. Once AEther Vial is in play, unless something lures you to hard cast a creature, it is unlikely that you would ever need to tap your land to cast your creatures again. Uncounterable creatures with flash is definitely not good for the blue mage.

Banned in Extended

A color specific example would be Genju of the Spires. At one mana, it would sneak into play almost 99% of the time and all the red mage has to do is to activate it enough and they would just win. Of course, this card belongs in the SB of a deck that has many Mountains in the deck. It is more of an extreme card than a good card because other classes of mages can simply trade creatures for lands and the red deck might just run out of resources to keep up.

Anti-counter tech #5: Play More Instant Speed Cards

Unfortunately for the blue mages, R&D has been brutal against them. Time Spiral block opened a container filled with creatures with a new keyword: Flash. Previously, without the keyword, flash creatures were probably just a handful. Simian Grunts from Urza's Legacy, King Cheetah, Raging Kavu and Mystic Snake are probably the few that existed a long time ago that I can name offhand (flash was good in memory magic many years back but I'm not very sure about now). After the keyword came out, flash became more "family" in the Magic world (which is to say, much more common). If you can play your spells at instant speed, you can make your opponent choose between countering that spell or the spell you might cast next turn. If you overload your opponent with choices, it's going to be a tough game for him with already more than enough choices to make based on the nature of the deck.

If a red deck manages to get enough damage across to put the opponent in a dangerous situation, every burn spell becomes a "counter this or die" card. You can eventually overcome your opponent by holding on to burn spells unloading them in two turns (the end of your opponent's turn and your during your own). If your burn spells can only be played at sorcery speed, it becomes more difficult to "overload" your opponent.

Anti-counter tech #6: MAN LANDS!

Before X ed, the man lands we had available were : Stalking Stones, Nantuko Monastery and Blinkmoth Nexus. In the new age, we have Mishra's Factory, the five Urza's Legacy man lands and the upcoming Mutavault. Man lands are the worst enemy of the Wogger (Wrath spammer). In standard, resolving Wrath of God or Damnation against any aggressive deck doesn't cut it as there are always more creatures raining down. To make things worse, man lands are just plain "unwoggable".  Man lands are "uncounterable" because they are not spells! If your deck allows, always try to put some in to make your deck slightly favorable against the blue mages you hate so much. (Just remember to activate them more often) Note: Svogthos, the Restless Tomb is another man land that fits in some decks (usually those with creatures in their yards). If I were to play it, I would be most likely to put it in the SB as a Living Wish target.

Anti-counter tech #7: Play Effects that are Not Spells

"Counter target spell" can only target a spell. By not playing spells, what you are doing is bringing the fight to another place where your opponent cannot react to.

Aeon Chronicler and Detritivore have the two best suspend effects that will mess up the blue mage. (Detritivore can only be matched by either having Willbender [and Vesuvan Shapeshifter ] in play or simply having basic lands.) 

Decree of Justice has been the finisher of choice in some decks because of its "uncounterable" nature. (Note: Do not tap more than three mana when cycling or you might get badly mana burnt to a Stifle or a similar effect) 

Urza's Factory is the land of choice used by most control players because it is both "uncounterable" and a token producer. Kjeldoran Outpost and Vitu-Ghazi, the City-Tree are both quite bad for the blue mage but they are less used nowadays because the 1/1s cannot match the 2/2 factory tokens very well and not many players would choose to play more than one of such lands.

Red players get Barbarian Ring, Keldon Megaliths and even Keldon Necropolis to punch through more damage. I personally do not like "come into play tapped" lands in aggressive decks because they lose more in the long run by coming into play tapped on crucial turns where the tempo loss becomes the gateway to the game loss.

Anti-counter tech #8: Discard Their Cards

If you start using Hymn to Tourach and Stupor and more discard, blue mages are going to counter those or start discarding counter magic to save their draw spells. Either way, you have the upper hand as long as you can drop something while they are open.  Spot discard like Duress or Thoughtseize pave your way through by allowing you to get rid of the most efficient spell they have in their hand and give you enough information to play into the opponent's hand. By being proactive, you force your opponent into making decisions at every point of your attack and the more decisions you are able to force your opponent to make, the more likely he will make a mistake you can capitalize on. Some people have trouble picking the "right" cards when they look into the player's hand. As a guide, try to pick the card that you think will be the card your opponent wants to cast. Usually when you watch games, some players let the spot discard get their cards while some do not. There are pros and cons to both situations. By countering the spot discard, players will be able to keep their hands confidential which might be very important especially if the hand is bad or there is something they do not want to show you. The bad part about this is that you lose the mana by casting the counter instead of letting him just take a counter. Also, try not to "loot" before a spot discard unless you think that you can nullify it confidently or you might end up getting the worse deal.

Turn 1 Hyppie is not good.

Anti-counter tech #9: Play Cards that Do Not Use Your Mana Now

Alternate casting cost, pacts etc. The hardest way for a blue mage to react is when he is not expecting something when it happens. Fireblast end games because you can continue throwing burn while being tapped out. 

In fact, if a person is "tapped out", most people immediately assume that there will not be any action at all from the opponent and will play accordingly (just like in my experience below).

Cards like these give you the edge because your opponent cannot accurately assess the state of the board or the game once you play with these cards that do not play according to the fundamental rule of the game: use mana to cast spells.

Sidetalk: I recently got tricked by an opponent with Urborg, Tomb of Yawgmoth hiding at the side while exposing a basic forest which I thought represented "tapped out for black mana". As my brain was fried from playing magic for the forth day straight (more than I've ever done before and getting little sleep because we stayed up late everyday - simply excuses for being bad at the game), I couldn't register the fact that there was the Urborg in play so I played accordingly to how I would have if the opponent didn't have black mana and I lost to Extirpate which shocked me so badly.

Anti-counter tech #10: Play Hate in the Sideboard

Post SB, if you dump in hate, it's going to be really really difficult for the blue mage to play.
Dosan the Falling Leaf and City of Solitude are the most extreme cards to accomplish this. The more color friendly Defense Grid might help in some matchups too.

If you need to get specific sorceries and instants to resolve, play Boseiju, Who Shelters All. 
Xantid Swarm, Orim's Chant and Abeyance are great hate for decks that need even more specific cards to resolve.

Usually, I would SB cards that fufill the objective of my deck while disrupting the blue mage. Cryoclasm is the best card (in my opinion) in this category as it attacks the blue mage from two angles.  

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Summary:

The key to winning against the counter matchup depends on how capable your deck is at outclassing the mana spending of the blue mage and how capable it is at winning them in the attrition war because counter magic are usually 1 for 1.  It is the best to play in such a way that stresses the opponent by giving him many options over a short, yet crucial, time span. This takes a big toll on the blue player and not everyone can make the best decisions under constant pressure.  

In other formats where cards come in playsets, people have the luxury of just using a small pool of cards for a specific purpose. In the singleton format, it is totally the opposite. To achieve a specific goal, one would have to look through different angles with a "diluted", or rather inconsistent, base to work with. These tips should allow you to construct a deck that would let you have a lot more game against counter mages.

Counter magic in Singleton have the same drawbacks of the format. There is only one Counterspell in the whole format. Every other piece of counter magic either has a different cost or has more text than the three words: counter target spell. This means that playing decks filled with counterspells are not as ideal as they can get. 

Nowadays, good decks make use of the powerful cards like Teferi, Mage of Zhalfir and loosen the emphasis on actual counter magic because of their inherent weaknesses and the large amount of strategies capable of engaging them. The only actual reason why I would want to put more counter magic in a deck would be to utilize Guile. *smile*

Last Tip of the Day:

If you want to hate counter magic while having a strong matchup against aggressive decks, I would suggest playing the URW combination because it allows you to play cards that are individually strong against both decks and can be altered to become a lot more powerful against each extreme post sideboard. However, you tend to lose out against combo decks so there probably is a balance.