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.
Dancing Chicken. Sometime in the late 1970s, the song acquired the name Vogeltanz (bird dance) or Vogerltanz (Little Bird Dance or Birdie Dance), although these names never caught on seriously in Germany. On some sheet music and recordings it is called Dance Little Bird..