As somebody who plays a fair amount of Singed:
Vlad, Rumble, and Teemo all do really well vs him. Vlad has the range advantage and out-sustains Singed. Rumble can keep Singed from engaging if you place your harpoons well and his shield mitigates most of the poison's damage, while his flamethrower means he can trade with Singed while chasing him. Teemo's sustained damage is greater than singed, and his kit allows Teemo to kite singed.
Overall, the best way to deal with him is to deny him farm. I was in a match a while ago vs Rumble, and he let me push up and farm. While I couldn't kill him, he didn't even try to kill me, and the game ended up with the enemy team failing to kill me 3v1 and surrendering shortly after. Harass or engage him constantly when he goes for farm, while keeping singed out of melee if possible so he can't e+q you. Your goal should be to force him to base constantly to heal, and the moment he leaves push up the lane as hard as you can so he loses the xp from your creeps.
If he's under tower, DON'T go for a kill, because he'll either run away or kite you around tower. Just continue to harass him so that he can't safely farm. Keep wards down and watch for enemy ganks, and you should be able to shut him down pretty effectively.
All this said, Singed either outright beats or forces a lot of champs into farm lanes, which is almost always in his favor. If it comes down to getting counter picked, beg for ganks and hope for the best.