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.
A male Raggiana bird of paradise does a dance to impress a female. When it comes to making some noise, birds of paradise are not shy! They call to announce their territory, to advertise their location to a potential mate, or to sound an alarm, but with different vocalizations, depending on the species.