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 mocking bird's ability to mimic, or mock the songs of other birds makes it a rare bird. It can sing more than 200 songs, mocking the sounds of not only other birds, but also insects, amphibians and even non-organic sounds made by machinery. The state's game bird is the bobtail quail.