Private Eye has nothing about singer (4)
I believe the answer is:
piaf
'singer' is the definition.
(I know that Piaf is a singer)
'private eye has nothing about' is the wordplay.
'private eye' becomes 'pi' (synonyms).
'has' means one lot of letters go next to another.
'nothing' becomes 'FA' (slang term - sweet Fanny Adams).
'about' says the letters should be written in reverse.
'fa' backwards is 'af'.
'pi'+'af'='PIAF'
(Other definitions for piaf that I've seen before include "Edith, the "Little Sparrow"" , "See 6 Down" , "Edith -, twentieth century French cabaret singer" , "French singer who regretted nothing" , "French singer, the Little Sparrow, d. 1963" .)