Perhaps the most well-known of them all:
Jingle Bells, Batman Smells
Robin laid an egg
Batmobile broke it's wheel
and the Joker got away
(there are plenty of minor variations - one that my little sister used to sing was "the Joker took ballet.")
As is so often the case, no one knows where the heck this came from, or when it started, though we can safely assume that it doesn't pre-date Spring, 1940, when Robin first swung into action (Joker was introduced around the same time). Some say it's been around since Christmas of 1940; the earliest versions I can find i print are from the 1960s (interestingly, the second line in most of the early printed versions is "Robin flew away." Since then, it's been all over, from The Simpsons to Hobie Hanson, You're Weird, a 1988 book by Jamie Gilson (who always had a way with working this sort of thing into her books). It's even in the title of a Junie B. Jones book.
How far back does it go? Anyone have a source on this from before the 1960s?