I use the X-large two door Havahart live trap. Armadillos don't see well and follow barriers such as walls, fences, etc. Place the trap along a barrier and use boards or concrete blocks or any material as wings to funnel them into the trap.
Every armadillo that comes into my yard eventually follows the walls to my large shop. One corner has an oak adjacent to it providing a pinch point. A little work with wings makes sure they all funnel through it and that is where I set the trap. It also catches non-target species such as the neighbors' cats, coon and possums. So far I've avoided skunks.
I use large tent pegs to pin the trap against the wall to keep the armadillo from moving the trap around. They will fight the trap and work their nose into corners and under the doors, bending them. They are extremely powerful and eventually one will escape by forcing a door open. Once you have to bend the doors and such back into position, you might as well invest in another trap as you will begin losing them regular. I find that this trap is good for 20-30 catches before replacement is necessary. At $80 bucks a pop or so, it can get expensive.
You need to remove them three miles or more or they will work themselves back, according to a number of studies on the subject. Squirrels too.