How do I play against Singed cheese?

You don't play against a singed, because he doesn't play against you. LOL.

Play Sion. Get Swifties. Ez pz cause your slow makes him unable to get to you before you can knock him up with a charged up Q.

MR?

Play a champion that scales well with free farming. Free farm because singed isn't going to be able to kill you if you avoid chasing him or try to kill him. Don't try to kill a singed, let singed kill himself, he will eventually. Since singed can't really deny you farm you will be able to just play farming simulator all game until you are big enough to just go stomp someone somewhere else on the map.

tl;dr The biggest mistake I see is people think they they need to kill singed which leads to them chasing and dying.

you proxy the enemy wave till you are unable to do anymore and still keep doing so since it doesn't really matter if you die, just keep proxying the wave and then waste as much time as possible of the enemy jungler and top laner. They'll more than likely lose so much cs to the turret while you are getting the entire wave with no issues due to nothing being there to prevent you from getting the cs.

That is pretty much the singed way so... just push wave, and then roam to mid and get advantages that way or help your jungler deal with the other jungler

Play a champion who scales well and has built in sustain with the option to buy more. Singed will end up shooting his team in the foot trying his stuff against a nasus, tryndamere, fiora, akali, irelia or anyone else who can itemize to sustain through the minion damage if he proxies and then scale up hard for later on.

Hell, just pick vlad and make him regret playing singed ever.