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 American Flamingo is the most widespread species of the flamingo family. 2  It is the state bird of Gujarat, India. 3  At the beginning of the nesting season, they perform mass courtship displays, where hundreds move together in a coordinated walk. 4  In ancient Rome, flamingo tongues were regarded as a delicacy.