...popular town runs from east Bury (5)
I believe the answer is:
inter
'bury' is the definition.
(to inter is to put into the ground)
'popular town runs from east' is the wordplay.
'popular' becomes 'in' ('in' can mean trendy).
'town runs' becomes 'ret' (I can't justify this - if you can you should give a lot more credence to this answer).
'from east' says the letters should be written in reverse (letters written right to left or 'east to west').
'ret' back-to-front is 'ter'.
'in'+'ter'='INTER'
Can you help me to learn more?
(Other definitions for inter that I've seen before include "Put under the earth" , "Bury a body - in the middle?" , "get put down!" , "foreign team" , "Bury - Milan football club" .)