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.
Sun Dance. The Sun Dance is an annual Plains Aboriginal cultural ceremony performed in honour of the sun, during which participants prove bravery by overcoming pain. The ceremony took place at midsummer when bands and tribes congregated at a predetermined location. The Sun Dance was forbidden under the Indian Act of 1885, but this ban was generally ignored and dropped from the Act in 1951. Some communities continue to celebrate the ceremony today.