Trendy fellow with a tailless rodent, notorious (8)
I believe the answer is:
infamous
'notorious' is the definition.
(I know that notorious can be written as infamous)
'trendy fellow with a tailless rodent' is the wordplay.
'trendy' becomes 'in' ('in' can mean fashionable).
'fellow' becomes 'f' (abbreviation - of a society etc.).
'with' means one lot of letters go next to another.
'tailless' means to remove the last letter (the tail of the word is removed).
'rodent' becomes 'mouse' (mouse is a kind of rodent).
'mouse' with its last letter taken off is 'mous'.
'in'+'f'+'a'+'mous'='INFAMOUS'
(Other definitions for infamous that I've seen before include "Notorious, having evil reputation" , "Morally bad, shocking" , "Heinous" , "Of evil repute" , "Notorious, having very bad reputation" .)