Two questions, what genre would you put stories about government espionage, secrecy and corruption encompassing things like the Bourne series, X-files (stretching it, but I feel like they were half in this genre) and Enemy of the State?

I'm writing a screenplay about the government imprisoning and hiding the existence of an AI researcher who makes a big discovery which happens to conflict with a secret government project. There are action aspects to it (his capture, and some trickery in slipping news of his whereabouts), legal drama aspects (the legal fight to release him once he is discovered), and sci fi aspects (AI).

Also, anyone else writing a government conspiracy or espionage related script?

It probably falls primarily under drama. Because that's the broadest genre you've got in there. And it kind of sounds like the action would be brief and not the focus. It could even go towards suspense with those kind of stakes. Or... drat I can't recall what the more specific name is for conspiracy/mystery/spy stuff.
But I couldn't say for sure without knowing more about the style and plot.

Yeah I guess its probably a drama. However, it does involve a lot of widely dispersed settings rather than a claustrophobic look at life in a secret prison or anything like that. Its more about how long it takes the outside world to figure out whats going on when the only connections my main character has is estranged parents, and a group of academic colleagues who live scattered across the world. And then later its about all the legal maneuvers and public protests, etc, so its got certain qualities of epics as well.

You might even come over to the Crime/Gangster/Thriller/Mystery Forum