I'd have to say I have the most trouble laning against a good Singed as melee. I don't really play ranged, but he can zone ridiculously well with his poison trail. You can't burst him down, so dives aren't really wise, and even by turrets, you can't really stop him from throwing poison trails down. If you try to fight him, he may just throw you farther out and kill you out there, or he may toss you to his teammates. I'm not sure he's broken, but he's tough to play against as Rengar, Shyvana, and Xin.
Lux is great too. She's got her slow and root and her Ult can really mess up a fight going your way from across the map. Like Singed, I'm not really sure that she's broken. But she's very tough to get a hold of to kill. You gap close to her, she roots you. You Flash or Ghost and she slows you. Or the other way around. It's tough to get close enough to finish her off a lot. And by the time she's kited you down, she ults you and it's over.
Darius is just broken in my opinion. True Damage that does scalable damage and refreshes on kill is just way too strong. I've seen Darius's wipe through teams so fast. Build up multiple bleeds, Ult, the next guy is low, Ult, the last guy is low, Ult. I really don't see how he's balanced at all in that regard. As others have mentioned, you can't afford to focus him either, because the other two on his team will own you. Darius is really tough to deal with. His bleeds make it worse, because even if you kill him, he may still get the kill (often enough to mention).
Jax has always struck me as overpowered. His dodge makes it virtually impossible to hit him while he's simultaneously destroying you. Like Darius, he gets stronger the longer you fight him. (Attack Speed) His stun on top of that just makes him impossible to deal with as Xin. The only thing you can do is knock him back, and often he'll just jump to you and stun you again. Jax hits ways too hard in my opinion. Just seriously WAY too hard. I wish I could tell you which items exactly, but since I've started playing (around when Renekton came out), I've never seen Jax hit so hard.
Tryndamere is worth mentioning. I know in the hands of the novice, he's nothing to be afraid of, but in the hands of a master, he's pretty much unstoppable. He slices through your team very quickly, has a heal, a gap closer, punishment for trying to run and a slow to boot, and invulnerability. When he's paired with Kayle, it's not even funny. Kayle herself I think is fine, but the combination of them together is very scary.
I do want to mention Teemo. Let's put it this way, any champion that can make you essentially "miss" your attack is a little over powered in my mind. Especially when said ability simultaneously does damage. Teemo is probably THE most used champion in TT because his shrooms offer map control, damage, and slow while he his extremely fast and hard. DoTs are nearly impossible to deal with and like Darius, even when you manage to kill them, they still kill you. Personally I think DoTs should be removed if a champion dies. Teemo and Darius do huge amounts of damage to you while they're alive, I don't see why they should get to do damage while they're dead as well.
Garen might be worth mentioning, but I don't really think he's broken, just a little "noob mode". He'sa cake walk to play and his silence is awesome. I don't really think he's broken though. He is a great champ on the map though.
Lee Sin and WuKong are probably the last two I'll mention. Wu Kong's ability to duplicate himself is beyond annoying and his ult, I feel, is like Death Lotus on a much tankier champ. His Ult is really a pain. Lee Sin's mobility is crazy and makes him very tough to lane against or close the gap to. He's got a way to move away from you, to you, and through walls. Just a little broken in my mind, but probably not enough to call him broken.
Those are my thoughts