I am trying to install a small metal media cabinet in a load-bearing wall in my basement, but unfortunately the cabinet is about 1/8" too wide to fit between the studs. Is it safe to shave off a small amount from one of the studs so that the cabinet fits?
First we don't really know if this wall is load bearing or not. If the house was built in the US 50 years ago and it is a basement with poured concrete then the chances of having a wooden fabricated load bearing wall are slim - it happens but not very likely.
Second all you have to do to answer your question - if it is a load bearing wall - is look above the middle of the two studs in question. If there is not a direct load point (pole or beam ending) between them above the two 2x4s then there is little/no consequence in making such a minimal move since the weight should be dispersed evenly through wall or at the very least past your 2x4s in question. If I were in your situation (and it were load bearing) I would simply tap one of the 2x4s over until I had room (shaving an installed 2x4 sounds like a lot more work than tapping one over) and then throw up a sister 2x4 on each side. If it weren't load bearing I would just tap one of the 2x4s over.