It's not "supported", but it might work. Not being supported simply means that it was not a consideration during design or testing of the server. Up to you to try unsupported scenarios.
The only "supported" Linux distros are RHEL and SUSE. You may have luck with others, but those are the only ones tested by Dell. Ubuntu has tested 14.08 and 18.08 on the R610, so they would be fine if you are looking for Linux … not that that includes only the SERVER releases of Ubuntu. Ubuntu does NOT test the Desktop release on servers, so again that falls to you to test.
If you are going to be doing mission critical services/data on it, then you'll want something supported. if it is not super important or there is some tolerance for downtime or random/occasional issues, then try Mint (I'm not saying you will experience downtime or issues, but you should not have the expectation that there will not be any).
All current Windows versions will work on this machine, as will ESXi (then you could run your Linux distros as VM's).
no role stated, say 1 HDD and simple file server of some kind, what is PURPOSE (ROLE)? 10,000 possible.
look on your dell support page R610 , been there yet and click OS tab?
what you see there is what Dell Support, (all)
so if not there , you are on your own
or go to mint forum , google that. and go there and ask there but you many find not answer but why not try it first.
The server will not catch fire, try it. see what it does, the dell support applications may not run at all or be flaky.
click OS here. tab
I just attempted to install Mint 19.1 (Cinnamon) on an R610 I have here, but it wouldn't even begin to load the installer. I tried twice, but both times got the same behavior; the installer did not even begin to load.
If you know you want a Linux flavor, I can say that I have successfully loaded versions of Cent, RHEL, Ubuntu, and Debian. SuSE has official support, so it should also work.
As for a recommendation, it really comes down to what you know and what the server's role is going to be. Hard to make any recommendations without knowing those.