I can't disagree with any of the posts here. I'll just reinforce that "it varies."
Literally, any unit can get ANY knife. It's just a matter of getting the commander (usually the first O3-O5 in the food chain) to hack off on it. Having said that, there are a number of knives "in the system." That is to say, they have National Stock Numbers", and the supply system can order them just like a pallet of sandbags, or a helicopter engine. Typically, they are considered Class II Expendable or Durable. It the case of the former, once it's issued, the army doesn't expect to see it back. In the case of the latter, it has to be signed for, and accounted for. The army wants it back when your done with it.
In most cases, the "durable" items are parts of kits and tool sets. Like the electrician's knife mentioned earlier in the post. They are required to complete these sets, and the next guy that has to sign for the set would expect the knife to be there.
Another example is an aviator's survival vest (I am an aviator). It actually containes two knives. One is either a multi bladed stainless steel pocket knife (cheap) or a standard leatherman (better). The other knife is a Camillus 5733 Pilot Survival Knife (the one with the brown leather handle). This is a really functional knife. I think you could bend the blade in half and not have it snap. The metal is so soft, you can sharpen it on your leather boots. This, of course, is preferable to a hard, potentially breakable knife. Of the two, one is Class IX expendable, and the other is Class II Durable. But the Army expects them both back when your done.
Enough on the Army supply system. On to the good stuff. I have been issued a BM AFO 9053 Auto. This is identical to the AFO 9050. The 9053 just identifies it as an Army Buy. If you go to the BM web site, you can find a section on GSA purchases. They have a non-auto folder, the 9053, and a fixed blade knife that they will sell to the government in bulk.
Other knives that I have seen issued are:
The MC-1 Para knife. An ironic side story here is that the auto part of this knife was supposed to be the hook. But none with auto hooks were ever produced. Only with the auto blade (about 3 in)
Of course, there are a variety of bayonets, new and old. My favorite which is the M9.
And a variety of cheapish utility knives, mostly for tool boxes. The BM 9053 is the only main stream, auto, primarily commercial production knife that I have seen actually "in the system." There are also a wide range of gerber and leatherman multi tools, as well as 3"-4" gerber folders.
As for the "rules", I have never been challenged on shipping or transporting a knife. I had to chuckle at my AF bother when he said "don't ask too many questions." This is very true. The official Army standard is "no longer than 3.5 inches" for personally owned knives. This applies to both fixed and folding knives. Having been stationed over seas for "peaceful" purposes, I have sent my various knives (nothing outragous) with the movers on a number occations. As for combat deployments, there is a wide margin of common sense. just about anything with-in reason is permitted. The only knife I've seen not permitted was a 5 inch survival knife with a spring in the handle and a release button that sent the blade part flying about 25 meters. I think it was called a "balistic knife".
Whew! Prolly more info than you needed.