Well, there was actually cardboard nailed up on the underside of the roof. 

Be especially mindful of any mold or moisture you see on the underside of the sheathing as this may indicate air leakage from the living quarters or a poorly ventilated attic. 

Appears new (as does the rest of the unit) and replacement of the carpet is unwarranted and I am mostly asking if a landlord in TN can charge only for a stain on underside of carpet with no damage to pile or paddingThanks in advance.

Putting it on the underside of the floor joists is possible, but not until everything is dried and the mold is taken care of where it won't continue to grow.

NOTE:  that subfloor was over an unfinished basement--we also used Kilz on the underside of the subfloor.  

Last a long time and are way quieter than PVC.Other things to be on the look out for:Mold, especially on the underside of the roof or exterior sheathing.

After several months of moisture build-up the kids putting in the sump pump found mold already appearing on the underside of the flooring & joists.

If the roof is sagging over a cathedral ceiling, it might be easier to inspect the underside of the roof from the outside opening the soffit.

The aerator is the small piece on the underside of the faucet stem that the water flows out of.

It appears that the mold is only confined to the underside of the particle board countertop around the sink edge.  2nd.