Brilliant band showing class in Sinatra number (5,3)
I believe the answer is:
milky way
'brilliant band' is the definition.
(a band of stars)
'class in sinatra number' is the wordplay.
'class' becomes 'ilk' (both can mean a type or variety).
'in' indicates putting letters inside.
'sinatra number' becomes 'My Way' (song by Frank Sinatra).
'ilk' placed within 'myway' is 'MILKY WAY'.
'showing' acts as a link.
(Other definitions for milky way that I've seen before include "Starry band of light across the night sky" , "Star-studded band" , "Lots of stars" , "Galaxy, broad faintly luminous band of stars" , "Millions of stars in a band of light across the night sky" .)