best wire fencing for coop run?

I use 2"x4" 12.5 gauge fencing for runs. I would never suggest that a person bury wire for protection. Most predator that dig can dig down a couple of feet without much trouble. You are better off to lay a foot or more of fencing along the ground on the outside of the run (predator guard), connected to your run fence. This will stop digging right were it starts, where the run fence meets the ground. They seldom figure out to move away from the run fence to dig. You do not have to bury the predator guard, just let grass or whatever grows there come up through the wire and it will self bury. Just pin the outside edge of the predator guards down with U shaped pieces of wire.

It depends on your predators.  Even though my chickens will be locked in their coop every night, I could never feel comfortable with holes larger than 1/2 inch because of snakes during the day.

Right..lay the wire, I could never dig deep enough through clay and rock...the coop is off the ground and floor is plywood...I did find a breech at the corners after buildind, a small gap I could put my finger through...enough for a snake or rat or mink, I plugged it, hammered in poplar plugs and nailed it in place---gotta be real picky when it comes to predators

If I ever build another one I will go with a walk in run 6 1/2 feet tall or taller made from chain link. I would then cover the bottom 3 feet with 1/2 inch hardware cloth. I think this is the most sturdy option for the best cost (if your buying new material versus used stuff)

My current run is left over field fence covered in 1 inch chicken wire. I have never had a problem, but a raccoon could fairly easily get in.

 

As already stated.... I think the predator guard style works great as a digging deterrent. The 1/2 hardware cloth would work great for this as well. 

Is your coop inside the run? You can use the same predator guards you decide to use for your run around the outside of the coop if it's not in the run.