Shame about the inspector (7)
I believe the answer is:
remorse
'shame' is the definition.
(thesaurus)
'about the inspector' is the wordplay.
'about' becomes 're' ('re' means regarding or about).
'the inspector' becomes 'morse' (Inspector Morse).
're'+'morse'='REMORSE'
(Other definitions for remorse that I've seen before include "Self-reproach" , "compassion" , "Regret or sorrow for wrong-doing" , "A feeling of deep regret" , "Guilt for a wrong deed" .)