Ashe is a fairly simple match-up for Varus. She simply does not do the same amount of damage you do, really at any phase of the game. The only thing you really have in common is a great late game ult that requires a bit of skill to use. Your key to winning this lane is to stand behind minions to avoid her Volley, and try not to get caught at level 1 with her passive 100% charges, as that crit will half health you in one hit. It isn't fun. So, i would suggest letting her hit that on a minion before you get in range for anything else.
Caitlyn is actually someone that you may think can outbully a Varus, BUT THIS IS WHERE YOU AARE WRONG. Her early poke is simply not a match for you, as you have good range, and even if she pushes you to tower, your W allows for you to have easy last hits. Should she go for an all-in, you out damage her, as long as you hit all of your spells. She will most likely use her net to esccape after she sees the damage your blight stacks do, so killing her may be a challenge. But after 6, your ult locking her down should do the trick. Should the Caitlyn be stupid aggressive and poke you all to hell, such as when she has a soraka support and has free mana, simply try to avoid as many peacemakers as possible and watch out for her level 6 if you're low.
Not many people really play Corki, but i assume if you went against one, his Valkyrie would be exceedingly annoying, and he'd probably bait your ult pretty often. He's pretty hard to catch, and his rocket poke after 6 can be pretty irrititating, but other than that, his damage is sub-par, especially with his super high mana costs for any kind of real poke. Whenever you run from him, assuming he has you low on health, try to zig zag so he doesnt hit as many rockets.
Whoever said that nobody can counter the Draven was WRONG. Carus is the biggest counter to Draven, not only for the fact that you can punish him horrendously for trying to catch an axe, seeing as you have an aiming reticule telling you exactly where to shoot, but you OUT DAMAGE THE DRAVEN. Maybe not at Level 1, but even as you hit 3, your damage is just better. And you have more range. He will undoubtedly push the lane, making it hard for him to catch axes, last hit, and harass. Meaning he'll be scattered while you can just farm away and get jungle ganks to shut him down right away.
Ezreal is a champion that rarely gives you trouble, but when he does, he gives you A LOT of trouble. Standing behind minions is the best way to counter him, as he cant hit Mystic Shots, but this may cause him to take a point in his W early to hit you. Luckily, it does no damage and costs a ton of mana, so he cant do it much. But you'll rarely kill him in lane, as his arcane shift is almost always up, and hes just so slippery. Honestly a skill-match up.
Many people say that Graves is the biggest counter to Varus. I've never quite understood why, as at the moment, Graves isn't very strong at all. He might do more burst damage than you, but your overall game presence is so much greater than his. Try not to feed him in lane, and try to deny him CS in order to make sure you scale into the beast you are, while Graves is stuck on the backburner, trying to catch up. Right now, Graves just isn't that much of a threat to any ADC.
Miss Fortune is arguably the biggest lane bully in the game, and one of the strongest ADC's right now. Her kit allows her to be super annoying, hit you for a ton, and still look good as hell doing it. She's fast no matter what, making skill shots hard to hit sometimes, and sshe does a lot of sustained damage. Your ults have similar uses, but she cant run out of yours. Pretty much try to deny her CS and dont give her kills. She's a monster late, and she can carry with even the slightest lead under her belt.
You will rarely see a Quinn in game, but on the off-chance you do, here are some tips. She hurts when she gets a harrier proc on you. She hurts when she gets a vault off on you. She hurts when shes a bird and can tower dive you pretty easy with jungler help, or even support help. Quinn hurts and is a snowballing beast. Dont give her kills early, and try to poke her a bit. What he lacks is an effective escape outside of an ult, minus the mediocre slow from vault, provided she can use it properly in the first place. This is another skill matchup, if you ask me.
She's really only a threat if she gets a soraka support, because then she can just spam all day. Her spell shield doesnt work well on you, and her mana costs are stupid high. Her level 3 Boomerang Blade can half health you early, so be careful of those. Try to get jungler help if she pushes, because msot sivirs who push wont stop.
I hate her. So much. She isn't really a huge lane problem, but if she gets kills, her late game is so strong. Personally, i think she scales better than any other AD carry in the game, with a crazy attack speed boost at level 18. She farms easily and cant help but push, so last hitting under tower is a must have skill.
Twitch is a strange champion. In the right hands, you will never win, but those hands are very rare. He doesnt do the damage you do unless he has a sona, and hes terribly easy to kill, at any stage of the game. His farming is sub par, and his only escape is a stealth which costs a lot of mana, a resource that he has a fragile supply of. Deny him cs and try to push, while keeping up ward coverage to avoid ganks.
Vayne is only dangerous if someone on your team is a next to a wall or she gets kills early. Both involve getting condemned, and both are not fun to deal with. She's a crazy late game threat, and if she reaches that late game by mid-game, your team has no chance. She can farm effectively at any stage of the game, and she can even farm champions when she has to, which is OFTEN DO YOU HEAR ME, SHE LOVES TO KILL YOU. Most Vayne players know they do a lot of damage, but think they do that damage all the time, which just isnt true. If she gets 6 hits off on you, your life is terrible, but as Varus, the amount of damage you do in 6 hits is comparable to her before she hits late game status. Dont feed her, dont lose. Simple as that.