Quiet about Conservative, popular, facing a scrap (9)
I believe the answer is:
scintilla
'scrap' is the definition.
(thesaurus)
'quiet about conservative popular facing a' is the wordplay.
'quiet' becomes 'still' (synonyms).
'about' means one lot of letters goes inside another.
'conservative' becomes 'C' (UK political abbreviation).
'popular' becomes 'in' ('in' can mean trendy).
'facing' says to put letters next to each other.
'c'+'in'='cin'
'still' enclosing 'cin' is 'scintill'.
'scintill'+'a'='SCINTILLA'
(Other definitions for scintilla that I've seen before include "flash" , "Slight amount" , "Tiniest bit" , "Tiny, scarcely detectable amount" , "littlest bit" .)