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.
Scarlet Tanager. Male Scarlet Tanagers are among the most blindingly gorgeous birds in an eastern forest in summer, with blood-red bodies set off by jet-black wings and tail. Theyâre also one of the most frustratingly hard to find as they stay high in the forest canopy singing rich, burry songs.