Birdshot will blow through walls, it can be fatal (generally infections days later), tends to cause sever disfigurement, but it doesn't penetrate enough to immediately stop an assailant. IOW, it will do everything you don't want it to, and nothing you do want it to. It is the biggest FAIL train in home defense mythology.
Birdshot will leave a nice whole in your wall (about first size with one shot, about dinner plate sized with 3-4) along with a nice spray pattern of little holes. I know this for a fact at least with #8 birdshot and my shotgun (Rem 870 - 28' smooth bore)
I thought the same (Birdshot doesn't penetrate) and shot at a target a few times quick fire (about 4 rounds in less than 4 seconds) and it destroyed the plywood the target was on. I actually felt bad as it was my friends range and I didn't know it would do that. One thing I did learn at least for me a pump is about 5x better than a automatic. A automatic to me seems to kick more, is heavier and followup shots are slower. With a pump I seem to be able to get back on target almost instantly by racking the pump.
As far as best rounds it's pretty simple that it goes about in the following order...
After that your really out of the Home Defense rounds and into stuff like #4 bird which might be ok, but I wouldn't trust it.
I keep 3 rounds of 00 Buck follower by 1 slug in my gun, after reading on here i'm considering keeping the gun loaded with 1 00 buck, with 4 00 bucks to follow. I would use 000 buck or #1 but it's hard to find and I know the pattern of 00 buck well in my gun which is my #2 concern only behind my safety.
Also I can confirm even a 380 will penetrate at least 1 wall with enough momentum to put about a 3/4 inch dent in steel. So even with a handgun you must think about shot placement and everything behind the target. For me that means I would have to aim low, or aim high which means it's even more important to know the pattern of your shotgun with the rounds you use.