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.
From the well-known national dance the Tinikling, which pays homage to the movements of a much-loved bird, to dances that reflect elements of daily Philippine life, these folk dances all offer a glimpse into the history of the country.