I'll see if I can answer this the best I can:
How often is the listen-server model used in the gaming industry?
When it comes to most online games, you'll find that a large majority of games use a client-server architecture, though not always in the way you think. Take any Source game, for instance. Most will use a standard client-server with a master server architecture (to list games available), in that one person will host a dedicated server and anyone with a client can join it.
However, you have some games and services, take for instance Left 4 Dead, League of Legends, and some XBox Live games, that take a slightly different approach. These all use a client-server architecture with a controlling server. The main idea here is that someone creates a dedicated server that isn't "running" any game. The controlling server will create a "lobby" of sorts, and when the game is started, the controlling server will add them to a queue, and when it is that lobby's turn, it will select a matching dedicated server (in terms of location/speed, availability, numerous factors), and assign the players to that server. Only then will the server actually "run" the game. It's the same idea, but a little simplified, as the client doesn't need to "pick" a server, only join a game and let the controlling server do the work.
Of course, the biggest client-server model is the MMO model, where one or many servers runs a persistent world that handles almost all data and logic. Some of the more famous games using this model are World of Warcraft, Everquest, anything like that.
So where does a listen server fit in here? To be honest, not really that well, however, you will still find many games using it. For instance, most Source games allow listen servers to be created, and many XBox Live games do (it's been a while, but I believe Counter Strike did, as well as Quake 4, and many others). In general though, they seem to be frowned upon due to the advantages of the client-server model, which brings us to our next point.
What are the (dis)advantages of it?
First and foremost: performance. In a client-server model, the client will handle local changes (such as input, graphics, sounds, etc) on each cycle of the game. At the end of the cycle, it will package up relevant data (such as, did the player move? If so, where to? Where is s/he looking now? Velocity? Did they shoot? If so, information on the bullet. Etc) and send that to the server for processing. The server will take this data and determine if every thing is valid such as, is the user moving in a way that indicates hacking (more on that later), is the move valid (anything in the way?), did the bullet from player 1 hit player 2?, and more. Then the server packages this up, and sends it to the clients, which then update whatever necessary, such as adjusting health if the player was shot, kicking the player if it is determined that they are hacking, etc.
A listen server, however, must deal with all of this at the same time. Since I assume you are familiar with programming, you probably realize how much power a game can rob from a computer, especially a poorly designed one. Adding on network processing, security processing, and more as well as the client's game, you can see where performance would take a serious hit, at least as far as just standard processing goes. Furthermore, most servers run on fast networks, and are servers designed to withstand network traffic. If a listen server's network is slow, the entire game will suffer.
Second security, as stated earlier, one of the main things a server will do is determine if a player is exploiting the game. You may have seen these as Punkbuster, VAC, etc. There are a very complicated set of rules that run these programs, for instance, determining the difference between a hacker, and just a very good player. It would be very bad for your game if you weren't able to catch hackers, but even worse if you executed action against a falsely accused one.
A listen server will generally not be able to handle the client's game, the server processing, and the hack detection, and in most cases, detectors like Punkbuster are very hard to, if not impossible to get to run on a listen server, because it's hard for it to function correctly without the necessary processing power, as generally the game logic is prioritized over security, and if the detector is not allowed to process for one frame it may lose the data it needed to convict someone.
Lastly, gameplay. The biggest thing about servers is that they are persistent, meaning that even if everyone leaves, the server will continue to run. This is useful if you have a popular server that doesn't have much activity in the night time, people can still join when they are ready to play and not have to wait for it to be brought back online.
In a listen server, the main disadvantage is that as soon as the client hosting the listen server leaves, the game must either be transferred to another player (creating a lul in the game that can last minutes in some cases), or must end completely. This is not preferable on a big server, as the host must either stay online (wasting a slot in the server, and his/her computer power, which could also slow the game), or end the game for everyone.
However, despite these problems, listen servers do have a few advantages.
Easy to set up: Most listen servers are nothing more than hitting "New game" and letting people join. This is easy for people who just want to play with their friends, and don't wish to have to try to find an empty dedicated server, or play with other people.
Good for testing: If one owns a dedicated server and wishes to change it's configuration, it is generally a better idea to test the configuration first. The user would either have to create a backup of the dedicated server and go blindly into the changes, with the only option being to roll back if something goes wrong, create a new dedicated server to test them, of just create a simple listen server to test them. And in with point 1, these are generally easier to start up and configure. This is especially true, as most dedicated servers are not within the administrators immediate access (most dedicated servers are rented from a remote location). It takes much longer to push configuration changes, as well as commands for restarting, etc, to a remote location than a machine that the administrator is currently on.
Less resources: In most dedicated servers, a user with the same IP cannot connect to the dedicated server (meaning, the client must either host the server, or play, they cannot do both). If the client wishes to play on his/her own server, they will usually need a second machine to host the server, or buy or rent a dedicated server so that they can actually play on it. A listen server requires only one machine, which may be the only thing the client can use.
In either case, both have advantages and disadvantages, and you need to weigh them with what you're willing to design and implement. From my experience, I believe that if you were to implement a listen server it would get used, if for nothing else than for a few users wishing to play around with friends, or test settings.
Lastly:
What other options do I have?
This is an industrial can of worms. In reality, any type of network architecture can be applied to video games. However, from what I've seen, like most internet communication, most boil down to some form of client-server model.
Please let me know if I didn't answer your question, or if you need something expanded, and I'll see what I can do.
