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 tikling bird was such a menace to rice fields in the Philippines, a dance was created after it. Honored as the Philippine national dance, tinikling â which means âbamboo danceâ â requires much stamina and concentration. Dancers hurdle between bamboo poles controlled by two people who slide and hit them to a beat.