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.
1 The Marbled Murrelet was once known as the Australian Bumble Bee by fishermen and as the fogbird or fog lark by loggers. 2  Though it was first described in 1789, a nest site wasn't discovered until 1961 by ornithologists in Asia; a North American nest was not found until 1974.verview. Marbled Murrelet: Small, chubby seabird with dark brown mottled upperparts and paler, white-tipped brown feathers on underparts, giving a wavy-barred appearance. Bill is dark. Legs and feet are brown. Feeds on fish and crustaceans. Swift direct flight low over the water on rapid wing beats.