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.
1 Several Northeast American Indian tribes including the Shawnee are known for a style of dance called Stomp Dance which is held for both social and religious purposes.  Although they did not always get along, the Cherokee Indians to the south were very important to the Shawnee.