So here are the jobs they do, per division:
No special duties have been noted for the 1st Division. The First Division is ranked highest among the 13 Divisions. Even subordinate members of the First Division are considered model Shinigami. Emergency must be dealt with swiftly; quick decisions and quick actions are essential. The division is able to ascertain a situation and mobilize quickly even before an order is issued. That is the true value of the First Division.
No special duties have been noted for the 2nd Division. Since the linking of the Onmitsukidō with the 2nd Division well over 110 years ago, its role may have changed from its original duty.
The 2nd Division has been directly linked to the Onmitsukidō for well over 110 years ago by former Captain Yoruichi Shihōin who was Commander-in-Chief of the Onmitsukidō before being promoted to a Captain. As such the special operations of Soul Society has become the full measure of what the division involves itself in primarily. Members are tasked with assassination, policing, detention and message couriers.
No special duties have been noted for the 3rd Division.
The 4th Division is the medical/supply division. They are responsible for both treating the injured and doing most of the manual labor, such as cleaning the Seireitei. The 4th Division is further divided into teams that have specific assigned jobs. Other division members use their spiritual power only for fighting. Members of the 4th Division possess spiritual powers that is used to tend to wounds and provide relief. They use these special powers and the benefits of their daily training to act with precision and speed to aid injured comrades. 4th Division administers emergency first aid to injured division members and transports them to the General Emergency Relief Station.
In addition, the nurses who work in the Sōgō Kyūgo Tsumesho (綜合救護詰所, the coordinated relief station) have special Shihakushō (死覇装, Garment of Dead Souls) that are pink in colour with hats that are reminicent of nurse uniforms of the Human World.
They are looked down upon by the 11th Division who usually bully them around because they believe that the 4th Division are just a weak medical team, as opposed to the 11th Division, which is combat-focused, although most members of the 11th Division are afraid of the Captain Retsu Unohana.
No special duties have been noted for the 5th Division. The members of this division are all highly skilled, having been trained by a captain who excels at everything, including battle readiness.
No special duties have been noted for the 6th Division. The 6th Division is seen as a model division by every Shinigami and is known for its strict adherence to rules.
No special duties have been noted for the seventh Division. This division is made up of sincere, unpretentious people who live life with gusto. Former Captain Komamura and Lieutenant Iba believe strongly in moral obligation and compassion. This ethos has been passed down through the ranks of the 7th Division.
No special duties have been noted for the eighth Division.
The mission of the Ninth Division is to protect the Seireitei. It is always on standby for combat because it is the security force of the Gotei 13. Traditionally, Ninth Division has overseen arts and culture. For this reason it is known to be in charge of the Seireitei news magazine. All the lieutenants of the divisions of Gotei 13 hand in their reports to them.
No special duties have been noted for the 10th Division.
The Eleventh Division is the direct-combat Division with a specialization in swords-only combat, forgoing the other Shinigami arts and thereby being one of the most specialized divisions in the Gotei 13. The division's doctrine of dominating in battle is what appeals to its members; they believe that fighting is what makes life worth living. This is what separates 11th Division from the others and makes them the strongest division. Everyone in the 11th Division are highly skilled in fighting. They believe if they are going to die anyway, go down fighting gloriously.
The division is also setup so that to become the Captain of the Eleventh, the candidate must do so through the third method of becoming a captain, killing the previous captain in front of 200 members of the Division, which is almost the entire Division. Upon defeating the previous Kenpachi, the new captain takes on the name of Kenpachi or the best swordsman of each generation. Since the rise to captain status of the current Kenpachi, an unofficial rule has been added that says to be a true member of the Eleventh Division, one's Zanpakutō must be a melee-type.
The 12th Division develops new technology and spiritual tools and its been the home of the S.R.D.I. for 110 years.
The 12th Division has been directly linked to the Shinigami Research and Development Institute over 100 years ago by former Captain Kisuke Urahara. As such technological and scientific research has become the full measure of what the division involves itself in primarily. Members are tasked with research and collection of data and samples, something both the current and previous Captains are very enthusiastic about, though not to the same extreme.
No special duties have been noted for the 13 Division.
Generally the kido heavy group. Also responsible for opening the Senkaimon, and the Sokyouku