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.
The Ê» elepaio is the first native bird to sing in the morning and the last to stop singing at night; apart from whistled and chattering contact and alarm calls, it is probably best known for its song, from which derives the common name: a pleasant and rather loud warble which sounds like e-le-PAI-o or ele-PAI-o.