For me, 38 Special is my minimum. I'm not a fan of .22LR as an SD round, simply because it's a rimfire cartridge and is prone to failure. The .22 Mag packs more punch, but suffers from the same rimfire problems. A pocket .380 or .38 j-frame is superior in every way to any .22 handgun in an SD environment.

Check out this link to an American Handgunner article on this very topic: .22 Mag Vs. .38 Special | American Handgunner