I think a pistol or a shotgun should be all you really need. There is certainly no reason not to have both however. If picking one or the other, a shotgun loaded with buckshot would be my first choice.
There are too many cons for me to consider a rifle a practical HD gun. This post sums it up nicely.
Rifle Cons: Overpenetration concerns; legal issues if engaging in long range SD shooting (this is kind of secondary); difficulty in moving around a house (opening / closing doors, etc with rifle in hand); easier for BG to grab from around a corner or behind cover
Never mind that two of the four listed cons exist equally with the shotgun, another is simply an issue of ammunition selection, and the other is a non-issue in most situations. A rifle carbine is no more difficult to maneuver around the house and no more likely to be snatched away from the user than a shotgun, over penetration is grossly exaggerated as a concern, namely because it can be largely eliminated with a little foresight given to ammunition selection, and rifles are good because they can engage at long distances if it is needed.
The point is that the rifle has the capability, which the others do not. In fact, the rifle has a lot of capabilities that the others do not--capacity, fire rate, the versatility afforded by changing penetrative capabilities with a simple mag swap, and lower recoil, to name a few.
Handguns are more compact and easier to maneuver. If I am just checking out a noise on the porch or a random sound in the driveway, I am probably just grabbing a handgun. But if I hear my door get bashed in or I hear strange voices in the living room, I am grabbing a rifle--probably my AK with a reflex sight. And truth be told, maneuvering with a carbine really isn't all that difficult, and neither is weapon retention. There is a position, which as I learned it is referred to as "Under Arm Assault," in which the butt of the rifle is pinched between the side and the elbow of the firing hand, with the axis of the bore parallel to the deck. In this position, the muzzle is naturally squared to the hips, which are squared on the target. The rifle can be held in this position almost indefinitely with little strain, and because the stock is tucked back, it serves to make the package shorter. In this position, one can use their support hand to open doors, turn on lights, operate a phone, handle a dog, guide a family member, grapple with an assailant, or whatever. Weapon retention is relatively simple as well, esp if the attacker is untrained and simply grabs the muzzle. A good sling doesn't hurt either. With the rifle slung, you can literally just let go of the rifle completely and let them play with the muzzle while you eye gouge them or punch them in the throat. It's a simple matter of training and common sense.
Of the handgun, the shotgun, and the rifle, the shotgun is the least useful and the most specialized. Compared to the capabilities offered by the other choices, its absence would be easiest to absorb. While both the shotgun and the rifle can handle the simple home break in scenarios, in a situation such as Katrina, the rifle's advantage in range, accuracy, capacity, and penetration could prove vital. Say a natural disaster strikes, power is out, and general lack of law and order ensues when authorities fail to bring basic services online fast enough. You have food, pure water, medical supplies, and batteries to keep you and yours safe, provided you can stay put and wait it out. It doesn't take all that much of a stretch of the imagination to have someone taking pot shots through your windows with a rifle from a couple hundred yards away. As noted, a rifle can easily penetrate several walls, and go most of the way through an entire house, if so loaded. With a shotgun, you are left no option other than to grab what you can carry and displace until the threat moves on of his own accord. With a rifle, you can at least return fire. If you're bugging out and someone engages you with a rifle, you can easily find yourself out gunned if you have only a shotgun or handgun. Not only in range, but in availability of cover. Neither the handgun nor the shotgun affords the ability to defeat barriers, if so loaded, to the degree of a rifle. And while these may not be concerns for the vast majority of home defense scenarios, in terms of usefulness and practicality, if only one can be chosen, the rifle wins, hands down.
Y'all are looking at it the wrong way. Rifles are not just long range affairs and when your life is on the line, there is no such things as "overkill." Rather, shotguns are simply close range affairs lacking the range, accuracy, and versatility of the rifle.