AR-10 has the best accuracy. If you are serious about deer hunting and want to scope your rifle, the AR-10 (and similar rifles like the SR-25) are by far the easiest to scope. There's a bewildering array of AR-10 rifles, from companies like DPMS, Armalite, Fulton Armory, etc. The AR-10 can be tuned to be a 1/4 MOA rifle. In tactical matches where semis are used, the AR-10 totally dominates.
The M1A is a great rifle as well. It doesn't have the 'assault rifle' cachet, and is legal in places like California. It is a good shooter, particularly in it's 'loaded' or 'match versions' - better than anything but a match AR-10. It is very rugged and reliable. It has the best iron sights of any military rifle. The trigger is great, and you can add a Rader trigger that is simply incredible.
Scoping the M1A requires a good mount. Avoid aluminum. Smith enterprises or Sadlak are the best, and they are expensive. Turning the M1A into a 1 MOA rifle can be an expensive proposition. however a loaded or match rifle will shoot on par with the better bolt action hunting rifles.
The FAL is probably the best of the 3 rifles in terms of ergonomics. It just feels good. They are reasonably accurate, though not equal to the previous rifles. Creating a super accurate FAL is problematic due to the design of the rifle, but it can be done. While there is a who range of accessories and gunsmiths devoted to accurizing the AR-10 and M1A, there are almost none for the FAL. The trigger is only average, and due to design, it is very difficult to improve. The iron sites on the FAL are very primitive, and the rifle was not designed for optics. Scope mounts for the FAL are less than ideal, and scope will end up being very high. This is a problem shared by the M1A, but at least in the M1A, the scope mount is secured to the receiver by one or two bolts. The FAL scope mounts replace the sheet metal dust cover that protects the bolt. Sadly, the best scope mount for the FAL is no longer in production.
The FAL does have the advantage of cheap magazine, but when you look at them, you understand why they are cheap. Still, they do work. The FAKL is a good choice if you want a basic battle rifle. While not as accurate as the other two rifles, the basic FAL can be had for as little as $1000, while the M1A ranges from $1500 on up, while an AR-10 starts at about $1200 for the basic model.
I've owned several AR-10 style rifles, M1As and FALs. I am currently using my M1A for deer hunting, as well as tactical shoots.