referring to a music band's song/ music

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.

11, for what it's worth (not much hereabout) as an erstwhile writer myself I'd strongly disagree with Con in this case

But I can't defend myself, I'm no grammarian. Doubtless we'll hear from others

I did write "I would... " at the start of my reply, to indicate that my personal preference would be to treat the name of band formed thus: the [plural word] as if it were a simple article+plural noun phrase thus:

the Rolling Stones are a rock band
the Beatles were a pop group

simply because it seems "natural", because I would say

the stars are bright
the dogs are barking
the books are expensive

However I am aware that there is a US vs. UK difference in the way plural nouns are treated, e.g. companies

General Motors makes automobiles
Rolls-Royce make cars

Your point, Con, is indeed well taken, meritorious. In the case of the OP however, the gap between the noun and its verb is large enough where the singular still sounds better (to me anyhow)

Ok. And what about a band whose name is a singular or an uncountable noun, or an adjective, or something else?
What variant would be correct in the example:
a) Deep Purple, a hard-rock band from Hertford, play their best known songs...
b) Deep Purple, a hard rock band from Hertford, plays its best known songs...
???