Most of the karaoke joints in the upstate, are very small roadside bar kinda places. Some are OK, some are real dives.
The biggest venue I've seen is called Backstage which is a nightclub on Anderson near Cherry, in Rock Hill. It's kinda smoky but draws a large crowd on weekends. They sing about half country and half other stuff.
Beef O Brady in Baxter Village on Friday is pretty good, sometimes crowded and sometimes just the regulars. Also take a look at John's Place or McHale's on Gold Hill Road on Saturdays, between Fort Mill and Tega Cay.
I've tried to get a karaoke club going for the area, I'll admit it's small - but I'd be glad to see some more people join up:
StateLine Karaoke (http://karaoke.meetup.com/373/ - broken link)