I have found half inch hardware cloth (a wire mesh) seems to work very well. It's a bit pricey, so I use it only on the night pen (Little Fort Knox). But we have all manner of predators and pests here. I also have two strands of equine tape (electric fence), one at about 4 inches above the ground, one at about a foot and a half. I use a Patriot fence charger attached to a battery.
That said, my runners only use LFK in spring summer and fall. Due to the fact that my runners never read the book about being winter hardy, they sleep at night in their own "room" in the walkout basement in the winter. They are outside all day right now (very mild days), and will be outside as much as possible once the cold comes in.
I like the idea of stacking the straw bales around the house for the muscovies. That's something I had considered before I discovered how much the runners were being run down by the cold at night.
What I'd like to emphasize is that protection from the cold and wet matters, and protection from predators matters, too.
To prevent digging under, I laid coated chain link fence across the bottom of the day pen, and half inch hardware cloth across the bottom of the porch of LFK. The footings under the porch go in about a foot on three sides, then I have an apron of a foot or two of concrete block and stone on the fourth side (too many roots and rocks to dig down deep enough).
