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.
Liturgical dance. Liturgical dance is a type of dance movement sometimes incorporated into liturgies or worship services as an expression of worship. Some liturgical dance had been common in ancient times or non-western settings, with precedents in the Hebrew religion back to accounts of dancing in the Old Testament. An example is the episode when King David danced before the Ark of the Covenant , but this dancing is often cited outside of Jewish norms and Rabbinic rituals prescribed at the time.