Pet given a separate resting place (8)
I believe the answer is:
catacomb
'resting place' is the definition.
(underground cemetery)
'pet given a separate' is the wordplay.
'pet' becomes 'cat' (cats are often kept as pets).
'given' is a charade indicator (letters next to each other).
'separate' becomes 'comb' (as in combing through something).
'cat'+'a'+'comb'='CATACOMB'
(Other definitions for catacomb that I've seen before include "crypt" , "underground vault" , "tunnel for burial" , "series of tunnels" , "Underground Roman burial place" .)