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.
Sacred occasions in Greek shrines, such as the games at Olympia from the 8th century BC, are inaugurated with dancing by the temple virgins. The choros is originally just such a dance, performed in a circle in honour of a god. In the 6th century it becomes the centrepiece of Greek theatre. In India the formalized hand movements of the priestesses in Hindu temples are described in documents from as early as the 1st century AD.