What are some of the peacetime duties of an Army Military Police Officer?

Writing people tickets for going 2 miles over the speed limit during PT hours.

The MP Corps as a whole are training for their wartime mission, Ensuring readiness of their units (property, training, medical, maintenance, etc), conducting Law Enforcement Operations, deploying as units, and some internment missions running military confinement facilities (not just the USDB at Leavenworth).

The LEO is going on at each Army Installation with MP units. They work with the Director of Emergency Services and answer to the Provost Marshall (a senior MP officer who may or may not be the Battalion Commander of the MP BN). The PM assists the Senior Commander maintain good order and discipline.

MP LTs are doing what their CO Commander tells them to do. Usually leading MP Platoons. Training, tracking LEO patrols, dealing with soldiers. Learning their craft as officers.

MP Captains are either BN, BDE, or DIV staff. They may be rotated through non MP units authorized MP officers such as Sustainment BDE Stafff. Staff Officers regardless of branch supervise their expertise area, work issues for units, track information, write plans and orders, conduct staff studies, and manages projects at the direction of the BN/BDE XO within the Commander's intent.

MP CPTs aspire to command MP Companies. They are responsible for everything the company does or fails to do. Company Commanders are responsible for the training, readiness, morale and welfare of the soldiers in their unit and their daily duties revolve around that.

MP Majors are staff officers, but most notably senior staff officers that are Provost Marshal, S3 Operations Officers, and BDE/BN Executive Officers (XO). S3 direct operations within the commander's intent, direct training, and resource training. One of the principle outputs is writing and publishing operations orders, plans, and fragmentary orders. The XO is the pinnacle. They direct the efforts of the staff primaries (S1, S2, S3, S4, S6, S8) and the special staff (Chaplain, SHARP, EO, Reenlistment Counselor, etc).

MP Captains and Majors may go to instruct at PME schools or serve as Service level, Joint, and Allied staff positions for broadening assignments.

Lieutenant Colonels are either senior staff at the General Officer level commands (Corps/Division/Combatant Commands) or in the institutional Army. Those selected, Command Battalions or selected Garrisons. Similar to what all branches LTCs do.

Colonels also service in similar staff assignments but are director level staff officers. Selected Colonels command Brigades and Army Garrisons.

There are a handful of GO positions ear marked for MP officers that become GOs. They culminate with the two star that is the Army's Provost Marshal who is the lead MP that advises the Chief of Staff of the Army. https://www.army.mil/opmg#org-about

Platoon leader, MP Duty Officer once in a while, additional duties (Armsroom OIC, Motor Officer, Voting Officer etc) you can also be selected to do 10% inventories that happen monthly.