To effectively rush with a shotgun in MW3 you have to have some common sense. You can't just run straight through the middle of the map expecting to kill people that are out of your range, and can see you running right for them. However, if there are multiple objects on your way to the enemies you can take shelter behind them whenever you are taking fire. You can also stick to the outer parts of the map, and pick off anyone that isn't sticking around with his team, or even kill multiple people by flanking them.
I haven't tried all the shotguns yet, but from the ones I have used I like the Usas-12, and then the Striker. The Usas-12 is a very effective shotgun as it has pretty decent damage, and pretty good range. The best setup for it would be having either grip or extended mags for attachment and either damage or range for the proficiency. I use Extreme Conditioning, Hardline, and Steady-Aim for the perks, and for kill streak I use Specialist and for the 1st unlock I use Sleight of Hand, then 2nd unlock is Assassin, then the final unlock is Scavenger. The reason I use Specialist is that I always find myself running out of ammo or always being killed because I run out of ammo or need to reload, but takes to long because I don't have SoH as my main perk, using Specialist reduces the chances of that to happen. For equipment I usually use Semtex and Flash Grenades, just so I can throw a Flash into a room if I know there's an enemy in there camping, or I can throw a Semtex across the map when I know my shotgun won't do any damage to the enemy at all.
For my Secondary I usually use a machine-pistol, only because it has some range to it, and I find it easy to switch to an Mp9 to finish off an opponent who is weak from me hitting him with my shotgun from a distance.
Tips for Shotgun:
- Do NOT run through middle of map expecting to kill the entire enemy team, they will see you and kill you before you get close enough
- Always know the range of your shotgun before shooting at enemies not looking at you.
- Try to stick around the boundaries of the map.
- You must ALWAYS be moving, if you stay in the same spot for to long people will catch on to where you're hiding at and wait for you to poke your head around the corner and kill you from a range since you're using a close-quarters weapon.
- Don't hesitate to pull out your Secondary to finish off an opponent or to even kill them from a distance with it.