Go with one of the first four live traps listed on this page:
http://www.bugspray.com/catalog/traps/armadillo_traps.html
These have proven to be very effective over the years. The kill traps can be implemented but only if you know where they are living and/or traveling. You’ll need such locations, like a passage tunnel or hole, where the trap will be placed to make the set. Without a good idea of where they’re active, it won’t be possible to use such a trap design. The Leghold design can be used if you have them eating and digging in your yard. The best size leghold would be either the # 11 or the #1.5. These traps need a central walking or trail area as well but will only need the animal to step on them and not pass through them like the kill trap so they can be more effective when used out in the open. These can all be seen in the Armadillo section here:
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