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.
El Jarabe Tapatio, a courting dance, has been popular since the  periodo virreal of Mexican history, and Felipa Lopez in Guadalajara choreographed the steps whichares performed today. The dance is distinguished by strong tapping and stamplng of the feet. Previously the men's and women's steps were different.