I have the exact same experience. It may even, unintentionally, be the _reason_ I don't listen to Pop music. It's generally not even a whole song - usually not even the chorus. I'll get a 2-4 second loop in my head of some insignificant section of a crappy song for 2 or 3 whole days.

Like you, it will sometimes start when I'm sleeping. I won't remember much of the dream, but the song is still there, echoing as it was in whatever setting my dream took place.