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.
ceremonial dance, ritual dance, ritual dancing-a dance that is part of a religious ritual. tap-a small metal plate that attaches to the toe or heel of a shoe (as in tap dancing). performing arts-arts or skills that require public performance.1. a piece of music suited in rhythm or style to a particular form of dancing. 12. a stylized pattern of movements performed by an animal, as a bird in a courtship display. da(u)ncer, Old French dancier, v. of uncertain orig.]. danceâ²aâ¢ble, adj.