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The Xbox 360 Controller for Windows is compatible on my Shield, but you'll need a USB port for the dongle.
Almost any bluetooth gamepad will work.
The GameSir G3S works. I was able to pair the DS4 controller once over Bluetooth (with lots of patience), but haven't tried again.
My Xbone S controller works great with Bluetooth, but using more than one at a time causes problems. I also use this keyboard with it.
xiaomi gamepad is aewsome and cheap
I can't really say for other consoles, but don't make the mistake I did and order one of the Mi gamepads from China like I did. It's just not a hundred percent compatible with the games and apps on the MiBox. If I had to do it over again I'd hunt for an original ASUS gamepad for the Nexus Player.