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 hummingbird has powerful spiritual significance. In the Andes of South America the hummingbird is a symbol of resurrection. It seems to die on cold nights, but comes back to life again at sunrise. Hummingbird is the creature that opens the heart.