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.
Other examples include: 1  Some mating dances, commonly carried out by birds, are examples of fixed action patterns. 2  In these cases, the sign stimulus is typically the presence of the opposite sex. 3  Kelp gull chicks are stimulated by a red spot on the mother's beak to peck at the spot, which induces regurgitation.