Popular dog to run up (5)
I believe the answer is:
incur
'run up' is the definition.
(I've seen this before)
'popular dog' is the wordplay.
'popular' becomes 'in'.
'dog' becomes 'cur' (cur is a kind of dog).
'in'+'cur'='INCUR'
'to' is the link.
(Other definitions for incur that I've seen before include "Fall liable to" , "Sustain" , "Bring upon oneself - be liable for" , "Arouse" , "Become subject to (something unwelcome)" .)