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The use of .410 bore shotguns for self- or home defense is a hot topic, currently, with the introduction of several handguns capable of handling ammo in that "caliber." The firearm forum that doesn't have a a bunch detailed discussions on this topic is RARE.
The comparison with 20 ga. shells is also a common topic.
MY PERSONAL preference is a 20 ga loaded with #2 buck. I own a "few" 12 ga, 16 ga, 20 ga and .410 bore shotguns (including a Mossberg 500E HOME SECURITY pump) - but the one that sits ready for use is a 20 ga.
If I owned ONLY a .410, I would choose one of the modern buckshot rounds loaded with 3 or 5 00 buck pellets.
Here's some info on 20 ga shot shells for defensive use:
And here's some info on .410 bore defensive shot shells:
Both of these articles are comprehensive and easy to understand. Read them until you understand what's being written.
DON'T consider "bird shot" - it is not effective, pellets are too small, don't have the mass and momentum to do much deep damage.