If you don't have any determined diggers 12" deep may be enough. Foxes will however HAPPILY and easily dig below that; potentially dogs or coyotes too if they really want in.
Another possibility to consider is putting your digproof barrier horizontal, on or just barely below the surface of the ground, rather than trenching it in vertically. Many people have good results with 2-3' of 2x4 welded wire used this way, either pinned well down or covered with turf or covered with rocks or concrete rubble or large pavers. (you would need hardwarecloth to stop rats, tho). The reason this works is that critters trying to dig in will typically start at the foot of the fence, rather than backing up 4' and trying to tunnel in from back there.
Any mesh or wire you bury needs to be a) galvanized and b) as heavy gauge as possible. Yes, even galvanized metal WILL rust underground (everywhere the coating gets scratched); how long it lasts is too dependant on climate, gauge, etc to make any general prescriptions about, other than heavy-gauge wire in a fairly dry climate can last "a good while".
If rats are a problem, I am not sure how practical it is to keep them out of the run altogether. Although it is true that hardwarecloth or closely-set large pavers (especially if buried vertically) would at least limit the amount of soil damage they do around the run fencing.
Good luck,
Pat