Indoor target practice at home is fairly common among Class III enthusiasts where I live. (Although the website I cited says you don't need a Class III for suppressors any more). I've not tried it; suppressors outdoors are quite pleasant to shoot without ear protection but you might still want EP indoors. The point of the suppressor is to avoid alarming the neighbors. It's also something I'd never try in an apartment or with other houses nearby because of the obvious danger if you miss the bullet trap.
I've been around a lot of suppressed arms, and aside from one competition rifle that was specifically designed to be silent (and cost many thousands of dollars to get that way), can't recall a single one in which ear protection wouldn't have been dramatically preferred for indoor shooting.
Even for that one (subsonic, small rounds, suppressed, nitro piston), it would probably have been irritating to fire indoors.
Edit: In hindsight, I suppose a suppressed, rimfire 22 would be pretty quiet.
I don't want to be a pedant: weapons are never "Class III" - this is just a tax designation for federal firearms licensees that are eligible to deal in NFA "Title II" items.
The distinction is pretty important, because you certainly do not need a Class III SOT to own a Title II item.
