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.
Considered as one of the oldest dances from the Philippines, this dance was originated in the islands of Leyte in the Visayan Islands. People of Leyte describe the tikling bird as one of the most unique in its movements - walking around and between the tree branches and some grass stems. This bird was named tikling from which the Tinikling dance got its name. Because of the creativeness of the people, they imitate this bird by using bamboo poles.