I was playing a melee against Singed yesterday, I was raging because I couldn't do shit to him, since I wasn't going to chase him around. So my friend who was playing Sona and I sat up top waiting for back-up, but it never came because said back-up had to keep helping out our terrible mid. :'(
Singed is fairly manageable levels 1-8, and very weak levels 1-3 (specially before he gets a catalyst), so poke him hard early. Worst he can do is either poison you (very weak early) or fling you once (long cooldown, but stay away from his turret). He has no ranged attacks, so use any ranged skills you have (and, in this case, Sona's autoattacks and offensive chords) to annoy him. Stay near your low-health creeps so you can punch him whenever he goes in for a last-hit.
Once his poison is up to level 3 or so, he might start killing your whole wave in seconds, and that'll bring him close to your turret, where you can farm safer and possibly gank him from behind. Don't hesitate to ask for a lane change if you know that your champion can't do crap to him, it's better to send someone else than to give him all the free farm. Once he's level 8 or 9, stop feeling safe under the tower, he can dive in, kill someone, and dive out.
Never chase Singed, that just makes him happy. Oh so very happy :3