What can I do once Singed has got enough HP that he can sit there taking my attacks all day? I usually play an ADC such as Tristana or Miss Fortune.
Build a Blade of the Ruined King for the percentage of current health damage, the active, and the lifesteal. Grab a Kitae's Bloodrazor for percentage of maximum health damage if he's majorly health stacking. A little magic resist wouldn't hurt either, to reduce his poison damage.
The most effective thing you can do is to not focus Singed in a team fight. In 1v1, if he's just running in circles around you, your lifesteal and the BotRK active should keep you alive long enough to wear him down. Don't freak out and stop attacking when he flips you, and be aware of possible assistance from his allies.
If he's actively running away from you, do not chase him. Even if you can dodge his poison trail, and you have a snare item, you're going to have a hard time attacking enough to wear him down. The further you chase him, the greater chance that you're running right into his allies.