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.
Though the primary threat to these sweet birds is habitat destruction. Possibly one of the most popular and well known birds that resides in the Amazon rainforest is the Scarlet Macaw, they are the largest parrot in the world and are very distinctive due to their bright red, blue, and yellow feathers.