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

Get a lawyer for free and your local news involved if it's off base. On base, there is already a checklist on where and what they are going to inspect.

Not only did I serve in the military but I grew up from birth to enlistment in the military. Granted that I have been out since 1983 but never have I known of any regulation allowing inspection of off base/post quarters. The only exception that I know of is during a joint local police/AFOSI,CID/NCIS/CGIS investigation. Unless you invite them in for possibly a welfare check or to confirm your landlord is violating rules, I see no reason for it to be acceptable. In fact, I would suggest you talk to your JAG and ask about it. After all, "Chain of Command" has been known to break the law in the past.

on post yes, off post--none of thier business, unless even if they are paying you to live off base such as S & Q sometimes they can get a little stupid in my opinion. in 1964 my wife and i lived in an old army nike missel site in the berkly hills of calif. and we were coast guard. they made me repair man for the site, but never once did they pull a stunt like that.

On a legal side your command can NOT enter any dwelling off post. Their Authority ends at the gate. If you are married or have roommates they can tell your command to go pound sand! You can also tell your command to go pound sand and call the Police. But… they can make your life a living hell. Personally I wouldn’t allow anyone into my house. I would contact JAG and my State representative. I would also talk to the local Law Enforcement Command and alert them to forced entry to Service Members homes which they will have something to say about do to the risk of hostility. When I went back to active duty after retiring from Law Enforcement I did put up with such shenanigans. And you shouldn’t!

AMAZING:
Military People Have Rights...
I Served From 06/29/61..(My 17th Birthday) - 06 /10/65; 19 Days Before Hitting 21,
And WE Had Rights Too.
The Right To #1.. "SHUT UP", And
#2..The Right To "GET BACK TO WORK".
Now It Appears To Be:
"The Powder Puff Championships, On The Air"
... WTF's Going On Now?..
This Is Just Plain-Azz "WEIRD"

Like you, I trusted my organization. When I was active duty I worked as the driver/courier of the DCSINT at NATO HQ, a 2-star general, the Army never asked to inspect my off-post living quarters and I had a TS SCI clearance. When I became a civilian with DIA and only needing a collateral secret security clearance, my supervisor said that all of our office were to allow the Inspector General to go through each room. I acquiesced but expressly stated that I wanted us all to stay together and not wander off. Only my supervisor disobeyed. I should have simply denied entry. There was zero reason for them to enter my private domain and there was no probable cause.

I was a Sr. Corpsman who had no authority but was expected to keep my eyes open. I'm glad I did. Just like your lieutenant and captain are to make sure your socks are good and feet in good shape its nice to know they care about your kid's socks. Unless you have a boyfriend like Barney Franks or your hydroponically growing herbs enjoy the fact someone gives a crap. Your officers wives are expected to help. Living away from family, your spouse away from mom, you out on orders and your family twisting in the wind, in the close family of military we give a shit. Reach out and have a pot luck Bar-B-Q. or small dinner with a few friends and your close officers and spouses or other. I miss the comrradery of my fellow military hurrying up to wait to take it in the shorts. I have horror stories of the spouse away from their support, the active gone on orders and her and the kids suffer because they took to long to ask. Just because you are active they are the members of a close fraternity and despite running into self important losers you will find those that are welcoming and helpful. Just like the rest of society you need to not give up.
You will get out of the military what you put into it.
I let my former wife talk me into getting out after quite a number of years. Amazing how many times I heard this after I was out. There should be a metal for sportive spouses. I know quite a few who deserve it to include a few that were Joint Chiefs of Staff. It's up to you.

They aren’t allowed to go inside. That is true. The only thing they’re really “inspecting” is making sure that you do in fact live in that house. There’s a couple of reasons for that-unit recalls, maybe a soldier doesn’t show for formation and their leadership has to go to their residence. After the fort hood incident-and all the new requirements for commanders regarding the missing soldier battle drill, finding out that a soldier gave the wrong address after they’re already missing and the Corps commander requires an update is bad bad bad.

Remember, we are talking the military...
You have the Rights, even Constitutional Rights, that they afford you. You are, even now, Government Issue, and they can abridge your Rights, using any number of scenarios.
Example(s)?
One is the fact that nobody has ever been brought to Justice for the Gross Medical Neglect of PFC Kenneth Lawrence Mosier (Ft. Hood, TX- March 15, 1979); The Base Commander's deflective answer about the death was that Kenny was under investigation in connection to an alleged auto theft. (His death was thanks to an accusation?)
Or, my Article 15 for disobeying an un-precise "legal" order. (Back story; soldier wife had a medical profile to wear maternity fatigues, due to a miscarriage and a new pregnancy. The Commander of RPAC-Europe, her Company, had ordered over the Easter Pay Day that the Uniforn Of The Day was to be Class A, thus going against a Medical Profile, when his mailroom/paymaster/CQ buck-sargent refused to release her mailed paycheck, and interfered with the USPS. I went, witnessed, and explained that he could answer to Senator Bumpers for his actions, then turned and started to leave. He then ordered "Private Mosier" to "come here!". Not being positive as to his addressing my wife, Pvt Mosier, or me, PFC Mosier, I turned and said that he needed to be more precise, as to the proper Rank. I was punished - He nor the Full Bird Kohler (Sp) ever faced any "Justice".
Your supposed Constitutional Rights are "full", as long as the Powers-That-Be grant them to you. And, yes, you are supposed to be better protected against such, without due process, but to circumvent the Fifth and Fourth Amendments, in these days and times, an "inspection" to verify that each servicemember does reside as they have stated, and to gain further access, I'd claim a "health & welfare check" and do those platoon/company/battalion-wide. Or better, sic the AP, SP, or MP, as need warrants.
Believe me, they know they're likely to have to ask for forgiveness, and face a slap on the wrist...

GOOD LOOKING OUT FOR OUR SERVICEMEN.