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.
In most traditions, birds have a predominantly positive connotation. Said St. Hildegard of Bingen in her Liber de Subtilitatum, Birds symbolize the power that helps people to speak reflectively and leads them to think out many things in advance before they take action. They represent the human desire to escape gravity, to reach the level of the angel.