Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
The Texas State... Ask any Texan, and you will no doubt learn that the mockingbird has the prettiest song of any bird native to North America. That's perhaps the chief reason the mocker was adopted as the state bird of Texas in 1927. The song of the mockingbird is, in fact, a medley of the calls of many other birds. Each imitation is repeated two or three times, then another song is started, all in rapid succession. In the above sample audio file, the songs of four distinct species were recorded in the span of about seven seconds.