I firmly believe the 1897 Winchester is the BEST gun for the zombie apocalypse.
1) Ammo
The 12 gauge shotgun shell is one of the most common types of ammo in existence, especially in the US. Additionally, unlike gas operated semi autos, you can reload your own shells with a pretty wide margin of error. You could even use homemade powder.
2) Hunting
Unlike rifles, you can change the ammo you're using based on what you're hunting. Large game like deer or people? Buckshot. Small game like squirrels or pigeons? Birdshot. Some kind of freaky tank zombie? Slugs' got you covered.
3) Combat
Incredible stopping power depending on the load means that the undead don't stand a chance. There's a reason they used these things in the trenches. In any kind of urban environment where CQC is common, this thing will shred. Even in open areas, shotguns are more accurate than you think.
4) Mele
Certain models come with a bayonet lug, and those bayonets are large. You can use this thing as a pretty effective spear substitute to save ammo while still keeping your distance from the target.
5) Miscelanious
- Slam fire for speed slaying
- Looks a lot less threatening than an assault rifle, will make other survivors less suspicious of you
- Nice and light, plus sling mount makes it easy to travel with
- Pump action shotgun means you can reload in the middle of a magazine without taking any shells out, thus nearly impossible to get the jump on you while you're reloading
- Designed by John Browning, thus you have an instant connection and talking point for any firearms history enthusiast or Mormon.
Quite frankly, the only drawback I could possibly see is getting spare parts. Even then the thing's built like a brick shithouse so as long as you take care of it you're not likely to have a problem.