Delicate stuff, oil turning up on cheese (7)
I believe the answer is:
taffeta
'delicate stuff' is the definition.
(type of delicate fabric)
'oil turning up on cheese' is the wordplay.
'oil' becomes 'fat'.
'turning up' is a reversal indicator.
'on' is a charade indicator (letters next to each other).
'cheese' becomes 'feta' (I've seen this before).
'fat' back-to-front is 'taf'.
'taf'+'feta'='TAFFETA'
(Other definitions for taffeta that I've seen before include "Lustrous silky fabric" , "Crisp, smooth, lustrous fabric" , "fine stuff indeed!" , "Fine crisp silk fabric" , "Stiff shiny material" .)