As of late morning, precipitation is spreading over the Upstate, but temperatures are mainly above freezing there, with the I-85 Corridor around 40°. Light rain has arrived over most of the Lowcountry.

A weather radar composite loop from 9:00 a.m. to 10:38 a.m. shows the progression of precipitation into South Carolina this morning.
Image Source: WeatherBELL
Precipitation will spread eastward and cover most of the state by early afternoon. Most of the state will see rain through the day, but areas along and north of I-85 will see snow or a mix of rain and snow.
Here's the regional breakdown:
Upstate and Catawba Region:
Judging by current weather radar and surface observations, areas south of I-85 in the Upstate will see mainly rain; perhaps some wet snowflakes will mix in. York County likely sees rain at first, then snow, while Chester and Lancaster County primarily sees rain. Parts of the I-85 Corridor had some rain at the onset, but it'll mostly be snow there. Areas to the north are seeing wet snow. With the back edge of precipitation already over North Georgia, precipitation won't last long. Snow lovers in this part of the state will get cheated by this happening during daylight in the third week of February. Temperatures are mainly above freezing, and strengthening February daylight will prevent much from accumulating. Shady areas might see a little slush on the grass and car tops; shaded road surfaces might also get slushy.
High spots along and north or west of Highway 11 will fare better with accumulations, with an elevation-dependent half-inch to two inches. You'll have to be on top of Caesars Head or Sassafras to have a chance to see two inches.
Precipitation ends later this afternoon in the Upstate and just after nightfall around I-77, and temperatures drop below freezing by late evening. Lingering slush or puddles on the road will freeze up, so icy patches will likely develop on the roads, possibly even in areas that see only rain. Watch out for 'black ice' patches if you're out overnight.
Temperatures may drop to freezing before the rain ends over the Catawba region (York, Chester, and Lancaster Counties), resulting in an hour or two of freezing rain. It might also get cold enough aloft that it just snows a little instead. Either way, there's a good chance for icy patches on the roads overnight in this area.
The expected cold front for Thursday morning will bring light snow or flurries starting around or just before daybreak. Most models show some areas getting a dusting, maybe even a half inch over the Catawba region. So, roads may be slippery, at least in spots, through midday Thursday.

Pee Dee:
This area will only see rain until nightfall when temperatures fall to near freezing. There remain some questions about how quickly it will get to freezing, how much of South Carolina will see freezing rain, and, in turn, how much ice buildup will occur. It won't be much, at most 0.10", so there isn't a risk for tree and power line damage. However, even a bit of ice can make the roads slicker than the proverbial greased pig. The primary concern is for elevated road surfaces to become icy. Precipitation hangs on until late evening in most places and perhaps after midnight right at the coast. Temperatures will drop into the upper 20s inland and near 30 at the coast later tonight, so icy patches will be around early Thursday. That may be true even for areas with no freezing rain due to frozen puddles here and there.
Thursday's cold front will bring a few flurries or a period of steady light snow to the region. Some models hint at a dusting over the upper part of the Pee Dee. Timing for this will likely be late morning and early to midafternoon.

Midlands:
I can't rule out the possibility of some wet snow mixing in with the rain late today, as it's coming to an end across the north, but it won't accumulate. There's a chance that temperatures will fall below freezing before the rain ends over the east, so an hour or two of freezing rain is possible. Temperatures in this area will drop into the 20s tonight, so even if no freezing rain occurs, icy patches are possible into Thursday morning as lingering water on the roads freezes.
Columbia metro and points south and west will see just a nasty cold rain this afternoon and early evening. Due to frozen puddles, an icy patch here and there can't be ruled out overnight and early Thursday, so be careful on the roads and sidewalks.
The cold front moving through Thursday morning and midday will likely cause scattered flurries. A dusting is possible locally in Fairfield, Kershaw and Lee Counties.
Lowcountry:
It's all rain here, with one possible exception. Before the rain quits tonight, there's a chance it will drop to freezing over a part of Berkeley and far northern Charleston Counties. Either way, that area will drop to freezing soon after the rain ends, so icy patches are possible over the inland northern Lowcountry late tonight and early Thursday. So, be careful driving in this area.
A stray shower or flurry can't be ruled out around midday or early afternoon Thursday as the cold front moves through.
Central Savannah River Area:
This area sees only rain this afternoon and evening. The cold front may cause a brief flurry or rain shower around midday Thursday.
Post-Storm Weather:
The extremely cold air mass over the Plains (lows were -20° to -40° again over Montana, the Dakotas, and parts of Minnesota this morning) will finally push eastward behind Thursday's front. Thursday night and Friday will be unpleasantly cold, with lows ranging from the mid-teens in the north to the mid-20s in the south, and temperatures only recover to the 40s Friday afternoon. However, temperatures will gradually moderate over the weekend.

Friday morning's forecast low temperatures from the National Weather Service
Image Source: WeatherBELL
There will be a chance for rain Sunday night into Monday as a storm passes to the south, with the best chance for rain over the Lowcountry.