Using an Epyc (7551) we have run the latest version that was available in November 2017 of the following distros. However, not all may run as you would want. For example,running a 32-bit version of ubuntu only uses 8 CPUs so you would not want to do that.
Mint 17.3 and 18.2
Ubuntu 14.04, 16.04, 17.04
Fedora 26 and 27
Debian 8 and 9 (+ 32 bit for Debian 9)
Any of the others should work too. We did see a few problems along the way. In testing for a problem of spontaneous rebooting under heavy load (see "spontaneous reboot" topic for details), We did find older kernels did not exhibit that problem. This is still being explored.