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.
The legendary roadrunner bird is famous for its distinctive appearance, its ability to eat rattlesnakes and its preference for scooting across the American deserts, as popularized in Warner Bros. cartoons. The roadrunner is a large, black-and-white, mottled ground bird with a distinctive head crest. It has strong feet, a long, white-tipped tail and an oversized bill. It ranges in length from 20 to 24 inches from the tip of its tail to the end of its beak.