I occasionally ride with, particularly if I know I'm going to have to change clothes sitting in my car or something. I do the same with running shorts, too.

And when I do wear undies, I have a handful of generic Dillards (Cabaret is their brand, I think?) that are made of essentially the same material as the insanely expensive patagonia underwear. They don't budge, they wick moisture, and they're seamless & don't chafe.