Bum stink can be used in many different ways. The reverse of the words "stink bum" to "bum stink" makes the use of the phrase a little more amusing.
"I am going to the toilet to do a Bum Stink" - Meaning "I am going to the toilet to do a shit"
"Have you done a bum stink?" - Meaning "Have you been for a shit?" or Have you farted?
It can also be used to tell someone that they have a smelly bum: -
"You have a really smelly Bum Stink" or "You have a really stinky Bum Stink".