I know vaguely when I should pick which adc, but I'm not entirely confident with when I should be picking what. And I'm not just referring to bot lane comps, I'm talking about in reference to your whole team comp, and your enemy's.

So can we compile a list of what team comps and bot duos compliment and counter each adc

Ashe:
Caitlyn:
Corki:
Draven:
Ezreal:
Graves:
Kog Maw:
Miss Fortune:
Sivir:
Urgot:
Varus:
Vayne:

Always Draven because Draven.

Mf goes well with most champs for kill lanes~!
I find volibear fun to have lanes with me when i play her, someone with a little cc to compliment her slow and speed so she can kill them quick :3
Sona works quite well and so does nunu, the champs that match mf also match vayne and other quick ad champs etc xD

Ashe: If your team lacks a good initiator or the jungler has no hard/soft cc for ganks.
Caitlyn: To outrange your opponent and poke your mirror lane to death.
Corki: When your mirror lane has no escapes, but the support has the ability to peel you off. Use R to secure kill.
Draven: For his strong early game, he becomes irrelevant late game unless you can catch his Q's.
Ezreal: poke, poke, poke, escape, global ult, as buff for team mates. Need I say more?
Graves: I honestly have no idea why people play this champ other than for his ability to take a beating and keep on kicking. Has a pretty good punch too.
Kog Maw: Low attack range, but due to his abilities he has one of the best late games of all of the ADC champions.
Miss Fortune: Great with an AoE team comp and probably one of my favorite carries in the game. She's a lot of fun and her ult range can be a pain in the ass to opposing teams.
Sivir: Spell Shield, Ult. Need i say more? A run speed and AS buff FOR YOUR ENTIRE TEAM? Too easy.
TWITCH: You forgot Twitch. Often overlooked and underestimated. If built/played right he can be DEVASTATING to the enemy team. True. Damage~
Urgot: You add Urgot... and leave Twitch out?! This champ... is not with the meta.
Varus: %HP damage and AP per hit? Seems pretty strong to me, not to mention one of the longest ranged pokes for ADC champs matched with a slow, AS buff, and a immobilization ultimate? Too easy.
Vayne: Vayne is so situational with team comp, lane comp and enemy team comps that its a hard call. She's definitely strong though.

Ashe: If you need kite and CC
Caitlyn: If you need kite and safety
Corki:If you need burst, poke and safety
Draven:If you want strong consistent damage
Ezreal: If you want burst, poke and safety
Graves: If you want burst and strong sustained damage
Kog Maw: If you want a tank killer and safety
Miss Fortune: If you want a powerful early game, poke and kiting
Sivir: If you want to counter burst, poke and strong sustained damage
Urgot: Never pick him now, nerfs were too harsh, has no role on a team anymore
Varus: If you want utility and poke
Vayne: If you want a strong late game

Honestly, Varus is my favorite ADC. He is also the only one that I have practiced enough to be decent with. I much prefer playing tanky bruisers, AP carries, or Junglers