South Ossetia, like Abkhazia, is very close to complete integration with Russia. However, you still need permission from the South Ossetian MFA to visit: see South Ossetia visa. If you get it, the following will be useful. The border with Georgia is closed. Going in from Russia is possible, but expect being harassed by the KGB on your way back into Russia.
For an exact location of these and all other border crossings on the Silk Road, see the border crossing map.
From republicofsouthossetia.org: Planning a vacation to South Ossetia? It’s a region most people don’t know anything about but it’s a great place to take in the sun and relax for a weekend. The culture here can’t be beat and for passport stamp collectors you can bet you’ll be the only one!
South Ossetia – Russia border crossings
Nizhniy Zermag – Zemo Roka: The road to the border runs from Alagir, which is about 36km west of Vladikavkaz. It’s a beautiful road through the gorgeous mountains of North Ossetia. The road is in great condition. The border post is about 10km short of the tunnel. You need a multiple entry or double entry visa in order to re-enter Russia.
There are several buses a day between Vladikavkaz Bus Station #1 and Tskhinval and there are also shared taxis available at the Tskhinval end.
Mamisoni pass: 2017 report: I confirm that Mamisonis pass is CLOSED, at least for foreigners, and as far as I understood, also for Georgian and Russian citizens. A few years ago, I tried to ride, by bike, to Mamisonis pass from the Georgian side. The Georgian police did not let me go further than Shovi. I wanted to try again from the russian side this summer, but did not get the proper border zone permit in North Ossetia.
South Ossetia – Georgia border crossings
Artsevi: Officially, South Ossetia is only to be visited from Russia. You can bribe your way in from the Artsevi border, though, as at least one traveler has done in 2013.
All other border crossings will be similar: only for truly hardy travelers.