Carbines are largely replacing both shotguns and sub guns. There is a reason for that. As one of the SWAT guys I train with once said "I only use a shotgun to breach doors or shoot dogs." This thread seems to be about "social" use, thus, I will forgo any discussions of obvious points that shotguns are better for wing shooting and the like.
I own, train with, use, and like both shotguns and carbines. For HD either it typically going to be up to the task if the user is.
When it comes to the type of use this thread is about I see shotguns only having two real advantages. 1) price and 2) terminal ballistics.
A carbine offers a number of advantages. Some of which have been mentioned above and some of which have not. The carbine is a smaller lighter weapon. This means it is easier for most people to use. I know a number of the 105-120 lbs girls I've dated had a much easier time effectively using any of my semi carbines versus any of my 12 gauge shotguns. This seems to have been a result of both size/weight as well as recoil.
A carbine has much less recoil. This makes it easier to use in a number of ways. One important way is follow up shots. If one has good trigger control they can put a number of rounds on target very quickly. Get a shot timer and fire 2-4 shots at target one, transition to target 2, repeat and transition to target 3. Do the same with a shotgun with slugs or buck, but only firing one round. This is also only accounting for semis. If you have a select fire weapon firing short bursts makes for a potent weapon. I know based on my time with the MP5, I would take that every time over a shotgun.
Ammo capacity: Carbines simply hold a lot more ammo than a shotgun. While I believe having more ammo than less is simply better (within reason), another issue here is that the limited capacity of a shotgun makes it more difficult to use. Proficient use of shotgun is largely about ammo management. Getting good at that takes a fair amount of training and practice. This issue is greatly amplified when comparing shorty carbines to shorty shotguns.
Over penetration and penetration of barriers: If we are talking rifle rounds, a properly selected rifle bullet poses less of a threat in these respects than a slug or even buckshot.
Precise shots: An advantage of a shotgun is that it puts 9(+) pellets out with one squeeze of the trigger. This also means one better know the pattern of their gun and be able to account for it. I have areas in my house that are large enough for the pattern of some of my cylinder bore shotguns to open enough that there would be stray pellets. Other guns with chokes and barrel work offer tighter patterns.
Suppression: It is much easier to effective suppress a carbine than a shotgun. Shotgun suppressors that are effective tend to be rather larger and heavy. Whereas a suppressor like an OPS INC M4M is effective and only adds 4" to the OAL. Suppressors offer a number of advantages for the type of use we are discussing.
Defeating Armor: A carbine in a rifle caliber is better able to defeat body armor. This may or may not matter in a given situation but nevertheless it is an advantage held by carbines.
The shotgun will win hands down on price every time, and I truly believe that is a large part of why they are as popular as they are for HD. If an MP5 cost $300 instead of $18K I bet you'd see a whole lot them for that role. Same idea for ARs and the like. If money is tight a shotgun is probably the way to go.
The terminal ballistics of shotguns is impressive. However, I do not find the terminal ballistics of carbines to be wanting. Given proper ammo selection even the 5.56 is formidable. If one wants they very same size guns are often available in 6.8 or the like.
These are just my random thoughts on the matter. I'd be okay having either, however, over the last few years I have really gravitated towards a carbine and my first choice. Beyond that, the fact that it is much easier for others to use than shotgun firing slug/buck makes it a better choice for my situation.
[shotguns] have a relatively simple manual of arms. They are also much easier to tote around
I'd argue that the manual of arms on many shotguns is more complicated than on a carbine. This is particularly true when you include managing different types of ammo, slugs and buck. Some people deal with that by using only one type of ammo, which negates versatility of the shotgun. Also, when they chose slugs only, I really wonder why they don't just use a carbine. In my experience it takes more work to become truly proficient in using a shotgun than a carbine. It also takes more practice to stay proficient.
They are also much easier to tote around
I'm not sure how a heavier and longer shotgun is "much easier" to tote around. If we include NFA items a 10.5" carbine, which is easy to haul around, is still really useful and sacrifices no ammo capacity. Most SBS guns that small give you a very low capacity. Furthermore, a 5 lbs carbine is still very easy for just about anyone to shoot. When you have a very short and thus light 12 gauge with defensive ammo that is not the case.