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.
Several kinds of beautifully colored birds of the Americas are called quetzals. The best known of these is the resplendent quetzal. The tail feathers of the male resplendent quetzal are about twice as long as the rest of the bird. The resplendent quetzal was held sacred by the ancient Aztec and Maya peoples.