Shed the last vestige of social class (5)
I believe the answer is:
caste
'social class' is the definition.
(caste is a kind of social class)
'shed the last vestige' is the wordplay.
'shed' becomes 'cast' (shed can mean to discard or cast off).
'the last' indicates one should take the final letters.
The last letter of 'vestige' is 'e'.
'cast'+'e'='CASTE'
'of' acts as a link.
(Other definitions for caste that I've seen before include "Large, long-necked, wading bird" , "Social position" , "Hereditary social class in Hindu society" , "place in society" , "Social status" .)