The normal route from the Georgian side is the Beknu route (Russian 5B, TD+/ED). I haven't climbed it but hope to this summer. The best topo I have found is on Summit Post here (3 images):
https://www.summitpost.org/shkhara-south-pillar-beknu-khergiani-route/92106...
Be aware that a lot of the info on that page is extremely outdated, Svaneti is very safe and there are many tourists (although there was an incident of mountaineers having all their gear stolen a couple of seasons ago).There is also a guide/topo here:
https://gnta.ge/publication/mountaineering-routes-of-georgia/
In my experience on other routes, these topos can be hard to follow as the routes are very long, but better than nothing.
I would also search the AAJ for routes by Archil Badriashvili or contact him on IG, he has a number of ascents on the south face (including in winter). He also has a few videos of Shkhara on YouTube. The reality is these routes see very few ascents compared to stuff in the Alps, so there is not as much information around, but it makes for good adventures.
