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.
Hummingbirds have long, tubular bills that resemble straws, which they use to sip nectar from flowers. Mergansers, specialized for eating fish, have sharp tooth-like structures on the edge of the bill to hold fish tightly. Hawks, owls, and other birds of prey which catch and kill live prey have sharp, hooked beaks. These are used to bite the skull or neck and also to tear the body into pieces small enough to swallow. The edges of a Mallard's bill are fringed to strain plants, seeds, and small animals from mud and water.