Personal Guard of the King, Police Within Wall Territory
The Military Police is the most prestigious due to their work allowing them to live within the Inner Wall. However, it is also the most corrupted military sector due a lack of oversight compared to the other sectors of the military. Only the top ten alumni from trainee squads can apply for the Military Police.
The Military Police is the police force within human territory. Not only do they maintain order, they serve as the king's guard. Since they do not normally come in contact with Titans, they must have special permission to use 3D gear and usually carry firearms instead. They are also in charge of tax collection and land distribution as seen in Ch. 15 when a couple policemen monitor farm production.
The Military Police is supposed to be an elite unit made of the most skilled soldiers due to its rigorous standards. The reality though, is a lazy, corrupt system that benefits from the powers granted to it. Though they are top ten graduates from the trainee squads; however, they are relatively the most inexperienced since they only stay in the confined safety of Wall Sina and never truly engaged in battle with the Titans before the incident at Stohess District where Eren and Annie engaged each-other in their Titan Forms.
The senior officers spend all their time gambling and drinking while leaving recruits to do all the work. They also pilfer taxes and misappropriate land as a recruit, Marlo, mentions in Ch. 31. This corruption is well-known with many people criticizing their lack of care for anyone other than themselves. Refugees and forced laborers (called producers) hold simmering hostility against them.
However, many trainees still dream of joining the Military Police because it will allow them to live within Wall Sina and safety. Also, being so close to the center of human territory, most policemen have relatively little experience fighting Titans as compared to the Scouting Legion or the Stationary Guard. This, together with corruption, loose discipline, contempt from public, makes this military branch the most incompetent out of the three.
However by Ch.45, it's revealed that the King becomes more tougher on the Military Police for their lack of diligence in dealing with their corruption, lack of discipline, pilfer taxes and mismanaging the lands. He orders them to leave the safety of Wall Sina and assist both the Scouting Legion and the Stationary Guards to look for the two other traitors in collaboration with Annie. By the end all three factions has banded together to help rescue Eren and Ymir from Reiner and Berthold.