Absolutely there are secret subways, Moscow and Washington are famous for them. It's an open secret that London is riddled with bunkers left over from the war that are still used by Her Majesty's Government today too. There is also a big question about their existence in Pyongyang and Beijing.

As for New York City, I highly doubt it. It's impossible to keep something like that a secret in that city. There are literally thousands of homeless people in the city that move in and out of the tunnel systems, they would have found it. The Metropolitan Transportation Authority is enormous, employs an army of workers and has volumes of maps regarding what is under NYC. Those maps are public and used extensively by urban explorers and historians to explore and document the system. There's just too much traffic under there for things to remain unknown. Even if there were secret tunnels, there would be an incredible number of rumors about them there, which there are not.

As for military subways in the western United States, it's highly unlikely that they exist to any great length. There could be subways between areas of certain large bases, that's possible, but the sheer size of the American west makes it a monumental undertaking. Think about how long it took to build the Chunnel or dig a new subway line in a major city, now multiply that by hundreds if not thousands of times. The manpower needed would be enough to employ the entire United States! On top of this, it has to be remembered that when you dig a hole that dirt has to go somewhere, entire mountains could be built out of a system that extensive, so where's the excavated material?