And they have to be reasonably well-known. My research has got me these:
Annie Get Your Gun - Squeeze
Annie, Let's Not Wait - Guillemots
Annie, I'm Not Your Daddy - Kid Creole and the Coconuts
... but there must be more, surely? Any help gratefully received.
UPDATE: crikey, that was more successful than I could reasonably have expected. Thank you blog people and Twitter chums.
UPDATE 2: this is also turning into quite a good 'misheard lyrics' thing as well.