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.
Duck dance. This dance is also known as the Chicken Dance, the Bird Dance, the Little Bird Dance, the Birdie Dance, Dance Little Bird, Tchip-Tchip, the Bird's Dance, Dance de Canards, Song of the Chicken, etc. SOURCE: Dick Oakes learned this dance from John Filcich who is one of the teachers who introduced this dance to folk dancers in the United States.