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.
Ghost Dance, central ritual of the messianic religion instituted in the late 19th cent. by a Paiute named Wovoka. The religion prophesied the peaceful end of the westward expansion of whites and a return of the land to the Native Americans.