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.
One of the best known of American birds is the robin. It nests from the limit of trees in northern Alaska and Canada to southern Mexico. Its musical warble, cheerily, cheer up, cheerily, cheerily, is one of the early signs of spring in the north. It seems to have little fear of people.