I sometimes bike with headphones, but only when I'm somewhere where it's pretty safe to do so (generous bike lanes, low traffic, non-technical, etc)

So... ones that revolve around my head are
Nirvana's "Teen Spirit" and "About a Girl"
Blind Melon- "No Rain" and "Paper Scratcher"
Smashing Pumpkins "Zero" and "1979" and "Today"

then some off-the-wall ones, like Snow Patrol "Hands Open"
and assorted Beatles, Grateful Dead "Stuck in the Middle", Jimi Hendrix, the Doors, and I'll randomly start humming 'innagaddadavida' only because it's the only song I know of that lasts about 15 minutes... or an hour if I keep forgetting where I am in it.