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.
Birds are powerful symbol metaphors for human moods and a range of emotions and beauty. The gentle Dove represents a feeling of devotion and divine, the crow refers to the nagging old woman, the flirtatious Partridge while the proud Cockerel represents amorous gentlemen.