My chain of command just announced that they're going to inspect on and off-post housing. What are they allow to inspect?

They can not enter your off post home. Even though some troops need the be checked up on.

A few years back my chain of command announced they were going to inspect our houses off post. I informed them that they were not authorized to enter my house and that if they did my wife would call the police and have them arrested for criminal trespass (which she would have). They backed down.

I dont think there intention is anymore then a health and welfare check. Sure as folks on here pointed out you can deny anyone you want entry into your home, but how do you think your chain of command would take that? I believe the purpose is just to make sure your doing OK.

Having been attached to AGI inspections, health and safety and building inspections in the military years ago, the only way your living on the economy housing could be entered is due to a complaint received by the inspecting department, weather it be by you or your landlord. And then the premise can only be entered with both parties present. And the complaint’s can be, for instance from the landlord, bad smell coming from the premises, high water usage, if the landlord pays the water every month. With residents, it can be almost anything that is complained about.
Somebody already said that most inspections are to
Insure the health and welfare of the troops that live both on base and on the economy.

Off Post housing does not fall under Military jurisdiction, so you are within your rights to decide that the only thing they can visually inspect is the front yard. If there is some form of evidence that the home is unsafe/unhealthy for the Soldier (whether the command has inspected the actual home or not) they can order the Soldier (i.e. you) to live in the barracks. If that happens, the Soldier does still receive BAH/BAS for their dependents. Good commands that do such will also have the soldier verify that the family is receiving the funds to pay rent and purchase food.

They cannot, it's not authorized, don't have authority to. They cannot inspect onpost housing if it's contract housing. Therefore off post falls in the same category.

Your off post, your house, they have no say or authority.

They cannot go to your house off post to inspect it without a warrant. ALSO, the military police have no jurisdiction off post. You cannot inspect a home off post as an NCO officer under any circumstances PERIOD. If this has been happening to you, you have a good case and can sue. It violates your rights under the 4th amendment; Amendment IV:
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.

For on base, I don't know the current rules/regs but OFF base, they CANNOT ENTER YOUR HOME WITHOUT A WARRANT. PERIOD.

I would REFUSE to let them in, either on-base or off-post housing. If they distrust people so as to intrude on their families home, something is terribly wrong with your chain of command.