I’m Canadian so I’m not that familiar with US military practices although I’m happy to tell you what I know.
I’ll start with Canada, because I know much more about that. In Canada, they don’t do much empty hand instruction. As a infantry sargeant once told me “Why would I need it? Most of the time I’ll be using my rifle, if I don’t have that I’ll use my sidearm, if I don’t have that I’ll use my knife, and if I don’t have that then I’m dead”. The modern battlefield is not empty hand friendly. That being said, he was taught some basic throws and strikes. Practically no kicks at all. They do teach a fair bit of knife fighting.
In Canada, Tae Kwon Do (TKD) is very popular on military bases, and last I knew, there was a Canadian Forces National Tae Kwon Do team. There is also some Kendo, Judo, and Aikido. A smattering of other disciplines as well.
In the USA, I did an informal poll once (1995) using students and their contacts in the US military. I don’t have the exact results anymore but one thing that did stand out was that all of the had some kind of martial arts club on base or associated to the base. Many of them were also TKD, 60% or so seems to ring a bell.
I do personally know a former Marine Corps hand-to-hand instructor. He taught Kempo in a civilian capacity, and he told me that they basically have their own curriculum within the Corps and it isn’t any particular martial art. I.e. it is similar to the Canadian military, basic throws and strikes.
I also know a former Navy SEAL. He told me that they teach a lot of knife fighting. For empty hand training throws, strikes and chokeholds are popular. The last being for eliminating somebody silently and without a knife. Again, no particular martial art, but rather what they want and what is effective from a variety.
I would assume that other military forces around the world are very similar, including other special forces in the US military.
Keep in mind, that all the techniques you’ll see coming out of the military tend to be very common, so you’ll see them in a variety of martial arts. Of course, the main introduction of martial arts to the USA came through military forces in Japan learning Shotokhan Karate (primarily, yes yes I know there were other sources), so ,as an educated guess, some of the techniques have their origins there.
One exception that really sticks out in my mind is Israel, where they teach Krav Maga; which, was also developed by the Israeli military. I know it is taught to their special forces, but I do not know if it is taught in general to their entire military. Israel is a bit of a special case though for three reasons:
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it is ALL urban combat, where hand fighting might be more useful
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combat takes place often with civilians present and so firearms might not be appropriate
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Israeli soldiers are heavily involved in policing actions, where again immediate lethal force might not be appropriate