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.
I have always loved dance â the sacred, the ethnic, the purely artistic and even the raise-the-roof, pump-up-the-jam variety. Movement is powerful and magical in all forms. But up until this week, I never really stopped to consider the Hula â the traditional dance of Polynesian culture.