I'd counter using a champ who does % HP dmg. Vi, Vayne, Fiora, Amumu (seriously...try this as counter to Singed...OP as hell), Fizz, Trundle and Shyvana are a few to try.
Here's the fun part - if you know you're going against a Singed, take Smite as one of your summs (personally I'd use Vi, Fizz, Trundle and Shyv, all of whom can get away with running Smite + Teleport top....they don't have to have Flash because of their kits). Buy hunter's machete and build it into Skirmisher's sabre (reduces Singed's dmg to you, your AA's will do some true dmg), then into Bloodrazor. That adds more % HP dmg.
Build Merc Treads, BOTRK, and then Hexdrinker/Maw. By now you've probably wrecked the Singed a few times...build whatever tank items your team needs, and you should be good to go.
Your AA's with the champs I mentioned should be absolutely shredding his HP, and he can't proxy for long while you're whittling his HP down because he'll eventually take harass from CS and towers....it shuts him down pretty hard. If you don't have experience with the champs I mentioned, you should spam them in games for a bit. I'd start out learning with Fizz or Trundle, and get some attack speed/lifesteal runes + masteries. They'll be the easiest to learn.