TEOTWAWKI Blog has a post about the viability of a single shot shotgun for preparedness/survival situations. You can click on the link to read his post on it. He makes some very good points, so please read his entire post before reading the rest of my little post.
He is making a few assumptions which while valid aren't applicable to everyone and every situation.
Here are a few reasons and the reasoning behind them that I use single shot shotguns.
#1: ease of use.
Reasoning: It is much easier for me to start instructing someone on how to safely shoot with a single shot firearm. There are less steps for them to try to remember, there are less distractions as well. When I have tried to instruct a new shooter with a pump or semi-automatic shotgun, they were much more leery of the various steps, plus many had a heightened expectation of just what those firearms are capable of. Many times they just wanted to shoot off all the rounds as fast as possible, ignoring the safety rules and trying to maintain a sight picture. With a single shot they are forced to slow down, to pay attention to what they are doing. There is less of a chance for them to try to fire as many rounds as they can as fast as possible.
Reasoning: If they aren't firing off as many rounds as they can, they can learn more about proper cheek weld and sight picture, even with a bead sight. The also learn how a proper hold reduces the felt recoil.
Reasoning: Many old single shots such as mine have no safety at all. I do have a newer Rossi single shot with a safety, however my old NEF doesn't have any safety at all. So drumming into their head that THEY are the first safety of any firearm can prevent future AD's and ND's. Teach them that the should not rely on any mechanical safety, but the safety in their head of safe handling procedures, and the 4 rules.
#4: Demystiying a firearm
Reasoning: Many people's only knowledge of firearms is what they see on TV and in the Movies. Showing them the actual mechanics of carrying, loading and firing one of the most simple firearms around, can do a lot about getting the reality of firearms into their forebrain versus the expectations of hollywood.
Reasoning: Many people, myself included, just enjoy the shooting of a single shot. Getting the best accuracy as possible with such limited sights, the feel of the stock during recoil, the satisfaction of a good shot hitting the target.
Those are some of my reasons for a single shot shotgun. In an actual situation, I do have other firearms available, but I would not feel "under-gunned" by having only a single shot. And for someone just getting started either in shooting or preparedness a single shot in good condition can fill many roles that they might need.