Mini-Ram hasn't had too many incidents though I remember a couple years ago when he was about 2 and had a really bad stomach bug. Wife and I both woke up around midnight (doesn't it always seem to happen in the middle of the night?) to poop streaming down his legs onto the plush carpet. It was a dual effort - wife cleaned the kid, I attempted to clean the carpet.
Now our biggest problem seems to be that he'll wait until we're in the car, on the way out/returning home, and that will be the precise time he needs to go poop. I have become quite the expert at finding restaurants/gas stations/ anything with a toilet.