You can look at a number of regular TVs which support FreeSync over HDMI. Otherwise, FreeSync and G-Sync support DisplayPort.
https://en.wikipedia.org/wiki/FreeSync includes a list of supported TVs/monitors, and GPUs/APUs.
https://en.wikipedia.org/wiki/G-Sync includes a big list of supported monitors and GPUs.
You can search up dedicated reviews for any that looks interesting to you, of course.
Unfortunately it's a bit hard to make universal recommendations because what you want is actually an "unusual" request to make of these monitors. Sure, FreeSync and G-Sync are both designed to work at 60Hz or even lower. But any LCD screen works better at high frame rates, with respect to motion blur / ghosting. (There are some methods to make 60Hz work at 120Hz but I'm not sure how good that is, and only RetroArch and GroovyMAME seem to have anything for this at the moment, or even good support for non-60Hz framerates.) At the same time, the better panels should be faster across the board. Likewise, not all that many monitors are known to have the really good input lag numbers of the latest high framerate gaming screens - and all this stuff adds up to hamper your quick response to what's onscreen. Sure, I've been using a high-end gaming screen, but I also use it a lot for 120/144Hz stuff, not just 60Hz things. It also cost a mint. I've been really happy with my combo - my GTX 960 was only around $150-160 after rebate, but the monitor was $800, and still is, and it's only 27".
There's a bit more to it, though. The main knock against FreeSync is the "FreeSync range." For LCD screens (not OLED), the amount of time it takes for a pixel to change is determined by voltage levels, called "overdrive." Apparently the appropriate overdrive amount changes when framerates change. G-Sync puts a module inside the display which, among other things, can change the overdrive setting so you get good-looking screens all the time. FreeSync's standard is simpler, and has to do with a set FreeSync "range." I've read a complaint about screen ghosting (though it's not clear if that's at low or high FPS). Additionally, if FPS drops below the FreeSync range, you probably get tearing.
The only thing that should matter for 60Hz is whether the overdrive setting is good enough. Otherwise, it seems that all FreeSync displays (minus one LG TV) support FreeSync at 48Hz or higher, and many go much lower, so you should always get smooth emulation of older consoles. Many FreeSync TVs will sync to 75Hz or so, though all do at least 60Hz. At this range you should get good performance (assuming overdrive isn't tweaked more for high framerates) and support for any classic console, so long as it's not (much?) over 60Hz.
