I would use the plural form (play) and 'their song' because the name of the group is a plural.
The Birdies, an experimental music group from London, play their song called...
Group names are typically treated as a plural if the name of the band starts with "the" and is a plural noun or based around a plural noun -- or more generally, if it seems to be referring to its members as a collection of entities -- and a singular otherwise. So we would have:
The Beatles are a great band.
Korn is a great band.
The Rolling Stones are a great band.
Metallica is a great band.
Led Zeppelin is a great band.