Well, considering his class background and that he first appeared in 1939, I'd think he probably listen to a lot of classical music. Though, I could maybe see him also listening to some of the more classically oriented big band like later Duke Ellington.
Superman...I can see him listening to a lot of Doris Day and the Ink Spots and that sort of smooth post-WW2 pop sound. Basically, the songs played on the GNR radio station from Fallout 3, but remove the R&B (so no Butcher Pete--but definitely "I'm as corny as Kansas in August").
Wonder Woman would have ancient Greek music like the Sieklos Fragment.
Ghost Rider would have Late 60s rock, like Born to Be Wild and Deep Purple.
Spiderman was introduced as a teenager in 1962...which means he's going to transition from Doo-Wop into British Invasion (Beatles) and Motown (Supremes)...he doesn't seem the type to listen Stax or Southern Soul...and I don't really see him going for the harder London Blues or Mod parts of the British invasion...so I don't see him into the Stones.