In my experience playing Singed (.. a lot) Illaoi is a pretty easy lane in my favor. Easy to throw you where you aren't a threat and such. And if I proxy you won't be able to walk between your turrets cause yeah no tentacles. And you have virtually nothing that keeps Singed in place so... Yeah.
Best I could say in that matchup is to have your jungler help, whether or not he proxies. Or just let him proxy. That's more or less how lanes that lose to Singed go, get jungler to help, if he doesn't, just don't die to Singed (he WILL tower dive you if he can kill you, so walk further back if necessary). If he Double/Triple Proxies, then he's either in over his head or just understands that your entire team lacks kill pressure on him in which case you lost to him at champ select. Hopefully it's the former in which case any ranged champ, even the support will do, can poke him until he leaves and potentially dies to turret. Ultimately Singed wants to get fed as fast as possible because his late game is really bad compared to his mid game, so simply stalling his gold income by getting him to leave lane a lot sets him back just as far as killing him.
Consider buying an Adaptive Helm early though, that cuts Singed's damage so freaking hard it actually tilts me.
Most other matchups though things are different. There aren't a lot of matchups in general where Singed comes out on top so he has to proxy to get farm; keep Jungler top side when dragon is dead and just kill him fast or chase him till he goes b. He's actively looking to waste 2+ people's time so even if you don't get kill gold you're better off ruining his proxy and calling it a day than you are pointlessly chasing him in circles.
