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.
In Hawaiian tradition, the Ê» elepaio was among the most celebrated of the birds. It is associated with a number of significant roles in culture and mythology. Chiefly, it helped kÄlai waÊ»a (canoe-builders) to select the right koa tree to use for their waÊ»a (canoe).