[@Raineh Daze] 9th division runs Seireitei News Magazine, on top of being declared the official protectors of Seireitei. Homeland security, basically (not that we saw that duty fulfilled a whole lot). 13th is specifically assigned to patrolling and protecting Karakura Town.

5th, 6th, 7th, 8th, and 10th have no immediately noticeable special duties, but they have their own flavor in that they are seen, respectively, as the well-rounded elite group, the model to look up to, the most moral and compassionate, pass on 8th and 10th though. Presumably just like the 13th and likely the others, they split up different portions of the world to patrol.