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 Tinikling. The Tinikling is considered by many to be the Philippines' national dance. The dance's movements imitate the movement of the tikling bird as it walks around through tall grass and between tree branches. People perform the dance using bamboo poles.