Our hardward cloth is outward about 12 inches and down about 3 inches, so far it has worked good. One of our dogs has tried to dig but stopped as soon as she hit the wire. We also have pavers around the gate to hopefully prevent any unwanted vistors from digging under the gate.
Anywhere between 0" (if you do it as an apron), in which case I'd bend it out at least 18" wide and preferably wider; to 18-24" deep if you do it as totally-vertical buried fence base.
Or anything in between - the more depth you bury it at, the more you can potentially 'cheat' on how much you bend it out, and vice versa.
Personally though I would not bother burying it, I'd just do it as an apron on the surface of the ground (pinned down well) or just under a layer of turf or rocks or pavers or mulch or whatever. FAR FAR easier, and just about as effective for everything except rats (and honestly not much will keep rats out)
I agree with Pat.
At first, I intended to bury my hardware mesh, but after thinking on it a while and seeing how hard it can rain here at times, I decided against digging a bunch of decent (albeit fairly sparse) grass and leave less compacted ground that is just asking to get muddy and erode easier.
Since I had a few logs around and a ton of rocks around my property, I thought I'd just rather lay down an apron and cover it in concerned areas with the large logs and the rocks. Also, I may cut some areas in the apron and plant bushes as well, or at least had put some thought to it. Once the bush got big, covered the ground, and the roots spread out, it may do a great job assisting in keeping the hardware mesh firmly in place.
For me, in PNW with my rain and weeds issue it was easier to bury so I could mow and weed wack without worrying about hitting it. Also the bob cat that tried to dig in actually started out almost a foot out so if I'd had a short apron he might have gotten in. I went 18" down and 12" out.
So far everything that's tried to get through the run fence has given up. We also had a coon try to dig in under the coop and come up through the floor. He bloodied his paws on the hardware cloth covering the floor, but he never touched a bird.
No burried hardware cloth for us either. We have an 18" skirt connected to the bottom board of the coop and anchored to the ground with landscape pins. This worked great. The pins hammered into the ground hold the apron tight to the ground. Grass has grown through the cloth and it's hard to see now.
We're opting for the apron here, digging anything involves pick axe and lots of hard labour due to nothing but rocks. It'll be hard for anything to dig it's way under for the same reason, so the apron is more to keep unwanted visitors from skinnying under the bottom rail.