Start to seriously rubbish American prestige (6)
I believe the answer is:
status
'prestige' is the definition.
(synonyms)
'start to seriously rubbish american' is the wordplay.
'start to' indicates taking the first letters.
'rubbish' becomes 'tat' (low-quality items or material).
'american' becomes 'us' (abbreviation for United States).
The initial letter of 'seriously' is 's'.
's'+'tat'+'us'='STATUS'
(Other definitions for status that I've seen before include "One's standing in the community" , "Importance, standing" , "legal position" , "Position eg in society" , "Relative social standing" .)