Disclaimer: Please note that these matchups are not guaranteed to win, but are generally good picks against their counterparts. Please also consider that not everyone plays the same and a counterpick for someone else, may not work so well for you. In fact, we all usually main 2 or 1 or 2 heros because we play those heros better than the rest and that is something to consider when picking too.

AD vs AD
Quinn counters Vayne
Trist counters Draven
Draven counters Sivir
Sivir counters Corki
Corki counters Ashe
Ashe counters Graves
Graves counters MF
Vayne counters EZ
Twitch counters Teemo
MF counters Twitch

Notables
-Caitlyn can do well in any lane and although she does have some popular lane matchups that she is picked into (believed to be a counterpick of) she is very ignorable if she isn't played hyper-aggressive and will sometimes become irrelevant the longer the game goes.

-Kennen falls off against everyone if built ad.

-Kog counters noone but outscales everyone at certain levels.

-MF can beat anyone in lane.

-Vayne's early game is actually quite powerful despite being squishy.

-EZ doesn't counter anyone, but soft counters Draven. It's because EZ can q where his axes are going to fall, but this is offset by Draven's ability to outtrade him and essentially out mid-game, Ezreal.

-Despite popular belief, Vayne can beat Caitlyn if played well.

-Varus doesn't counter anyone, but lanes really well against everyone. You pick him for his laning phase.

Picking Your Support

Generally speaking, there are no real "counter picks" it's just simply a matter of what is commonly accepted as good.

If you can theory craft it, then it can be good. For instance: you are against an MF and Sona bot lane.

A good example of some bottom lanes you want to run against that would be :

draven, leona
leona, ez
taric, caitlyn
janna, vayne
alistar, twitch

You just want to make sure that there is some kind of synergy between the two heros. It's not like there is any one combo that is the best, but rather each combo is dynamic and has certain strengths and weaknesses which will be personified by each persons skill level.

An example of a bad lane to pick against mf sona is something with less sustain than sona that has no hard engage.

If you are picking against MF purely, you would pick something that can lane against her early game and scale just as well as her in every part of the game like Varus.

Since you also have to account for Sona in this matchup (as well as your own support and potentially their entire team), it may be a mistake to pick Varus without a sustain support.

The tradeoff for picking a non sustain support is that you will either be outlaned completely or slightly outlaned for the chance at total domination of the lane and glorious glorious kills.

There are exceptions to the non sustain support in terms of aggressiveness (tTaric, Alistar, Sona, and in some ways Nami), but generally speaking the supports with the strongest laning phase all ins do not carry heals in their kit (Leona, Lulu, Janna, Lux, Nunu).

It's pretty insignificant to classify heros by whether or not they have a basic ability that heals in relation to how they are played, but the correlation is undeniable. Non sustain supports typically have the better all in during lane, but have a weaker laning phase in general which they trade for a better late game.

Conclusion

There are plenty of lanes that go even against each other and because there is always another person to consider in the lane, knowing how a certain combination works versus another is pretty important.

You will only learn this with time and it's something that's different for everyone even if there exists popular counter picks.

Learning what works for you and not trying to figure out what everyone else is doing is probably the best thing you could do for yourself. At the same time, you should also consider what's popular because it is most likely popular for a reason. At the very least it is your job to know why. Thank You