Shotguns are great for HD. But I won't leave them out for kids or the intruders themselves to find, and no, I'm not carrying in my own home.
The algebra of invasion was figured out a long time ago, and that it's the invaders have the advantage because defenders have to figure out how to defend against every kind of attack, and the invader only needs one that's successful. Do the math and ask if in one of the safest countries in the world if you are genuinely at risk.
I don't want to live in a compound like some goony prepper, so I don't worry about home invasion much if at all. Besides, at my house even the Fed Ex guy can't find my front door.
(edit because I hit the post button by accident) I use my hunting guns for HD as much as I think I won't need them. Mainly because I know exactly what they can do. As far as loads go? Bird shot stays outside, because I'm too lazy to bring it indoors, but no need to drive 00 through the walls. So somewhere between that works just fine. No one wants to get rat holed by a shotgun no matter what the shot size.