Heavy-gauge chainlink or wire mesh (no larger than 2x4") will generally be strong enough to stand up to anything short of bears, assuming it is well attached to well-set posts. On the bottom 3' or so, you can back it with hardwarecloth or 1/2" chickenwire to prevent things from reaching through, and to make it so that if weasels want in they will have to climb to do so.
For digproofing, one approach is to bury your (heavy gauge, galvanized) mesh fence a couple feet into the ground. (Don't try this with plywood, it will rot through REAL FAST, possibly not quite as fast in a dry climate but still not a good bet there compared to your other options - worst of all, you will not know how rotten it has gotten).
This is quite a lot of work however, and buried wire WILL start rusting through and weakening, yes even if it is painted or galvanized, although the heavier the gauge the longer it'll take to become flimsy enough to admit predators. Seems a shame to put all that work digging a fence in when it will not last forever, be a huge amount of work to replace, and you can't really monitor its condition).
Thus personally I am a bigger fan of the other main approach, that being a wire apron stretched out horizontally ON the ground (or just beneath the turf if you prefer) for 2-4 feet out from the base of the fence. Again, use heavy gauge wire mesh no larger than 2x4". You can pin it down well, or weight it with rocks or concrete rubble or pavers, or bury it under turf, or whatever you please. Predators will do their digging at the base of the fence, encounter the mesh, and be unable to proceed. Only a rare few animals (such as a few very educated foxes) will ever figure out to start digging far away from the base of the fence. Of course your wire mesh will gradually rust, same as if it were buried, but you didn't have to do very much work to install the apron and can easily check its condition and replace when needed.
As far as the top of the fence, the safest thing is an actual roof, engineered to withstand your winds and snow load. That is expensive and may involve building permits, but will absolutely keep out climbing and flying predators. If a roof is not an option, you can use wire mesh (preferably no larger than 2x4, preferably NOT chickenwire which raccoons can rip apart) but it will need MANY STRONG supports, almost as if it were a solid roof, to resist snow (yes, snow WILL accumulate on a wire roof
). That's pretty predatorproof too. Third place would be netting on top of the run, but that will only protect vs hawks and keep chickens from flying out -- raccoons, possums, foxes, dogs etc that climb the fence can go right thru the netting. Also snow will take it down.
You can beef a fence up with a couple strands of electric wire, but I would really suggest not making electric a mainstay of your predatorproofing because sometimes it WILL go out and animals can be quick to take advantage. It can be useful as a little extra insurance in some circumstances however.
Good luck, condolences on your loss,
Pat
