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.
What's sad is, since they cost only a few bucks at a pet shop, they don't get the respect they deserve.. And like all parrots -- all birds, really -- budgies love music and, like the famous Frosty the Cockatoo, are likely to break out in dance!