There is no rule that says a HD shotgun has to be styled after the military or police or even be tactical. Like Minnow said above, it just has to be loaded and ready for work.
It could be your hunting shotgun. Would there be some disadvantages? Sure but it will still blow a huge hole in a bad guy if needed since most home defense situations involve falling back to a secure room, lock the door and pointing the shotgun (or any gun) at the fatal funnel until there is a target to shoot at.
Quietly clearing your house like a SWAT team member sounds fun but in most cases isnt realistic especially if its your wife whos home alone doing the defending while you are working swings or graves or just not there to be Joe SWAT.
If shopping for a dedicated HD shotgun, then like previously mention above, keep it short i.e. 18.5 to 20 inches usually. Extended mag tubes are useful but by no means a must. I'd stick with a traditional stock (no pistol grips unless it has a stock on it too). If you have the cash, a simple white light and a way to carry a few extra rounds and that is it. Why? So you can have a complete package (shotgun,ammo and light) all in one grab late at night. You wont forget to grab these things if its already on your gun.
It could look like this if you are serious into shotguns
Or look like this if you are on a budget?
(This last one was a joke btw...not my pic but I thought it was hilarious the first time I saw it)