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.
Birds claim territory through a number of behaviors, including: 1  Singing: Singing is one of the most common ways birds advertise that a territory belongs to them. 2  Songs will carry quite far, and birds will perch near the edge of their territory to broadcast their claim to the maximum range.