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.
This word of Swedish derivation, describes a pattern of mating behavior seen only in a small number of birds around the world. In North America its chief practitioners are Greater and Lesser Prairie-Chickens, Sharp-tailed Grouse, Sage Grouse, and Buff-breasted Sandpipers. During the mating season, the males of lekking species gather into small clusters of territories, called leks, or arenas. Each male defends a territory within the lek although the area may only be a few yards across.