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.
Courtship display and mating: When a female enters his territory, the male performs a courtship display. The Ruby-throated Hummingbird male will flash his red throat feathers and perform a dive display, rising 8 â 10 feet (2.5 â 3 m) above and 5 â 6 feet (1.5 â 1.8 m) to each side of the female in looping dives.