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.
Not to be confused with The Chicken (dance). The Chicken Dance, also known as the Birdie Song or the Chicken Song, is an oom-pah song and its associated fad dance is now a contemporary dance throughout the Western world. The song was composed by accordion (Handharmonika) player Werner Thomas from Davos, Switzerland, in the 1950s.