Several answers, and a few comments:
1) What are you trained, practiced, and proficient with? Do you put several boxes of shells a month through a shotgun in defensive shooting drills, or several boxes of ammo through your handgun each month in defensive drills? Which gun will you shoot the best, the fastest in the kinds of close, fast, "practical" shooting that might be required to save your life?
2) Which will you have with you the moment you need it? Having a gun in the closet down the hall back in your bedroom, when you need one at the front door, the back door, or in the garage is exactly the same as having no gun at all.
3) Birdshot is for birds. Stick with No.4 buck or larger for home defense. Reasons for this are well established and available elsewhere -- do a search.
4) Knock off the "searching the house" or "investigating a noise" silliness. Study the S&T&T forum here (or much that is written elsewhere) and learn how to defend without painting a target on yourself by going on the offensive. "Investigating a noise" means one thing: You've already decided the noise ISN'T anything dangerous or needing your attention because if it WAS a threat, you'd be at a lethal disadvantage the moment you stumble into its view/range. Smart home defense involves hunkering down in a defensive strong-point and using your eyes, ears, patience, and other tools (like the dog!) to inform you and keep you and yours alive. (Search on "Don't clear your house.")