Trainee detective comes up without notice (5)
I believe the answer is:
cadet
'trainee' is the definition.
(cadet is a kind of trainee)
'detective comes up without notice' is the wordplay.
'detective' becomes 'tec' ('tec' can be a synonym of 'detective'**).
'comes up' shows that the letters should be reversed in order.
'without' means one lot of letters goes inside another ('without' can be similar in meaning to 'outside').
'notice' becomes 'ad'.
'tec' reversed gives 'cet'.
'cet' placed around 'ad' is 'CADET'.
(Other definitions for cadet that I've seen before include "Apprentice officer" , "Trainee -- acted (anag)" , "novice" , "Trainee soldier" , "Officer in training" .)