In keeping with the idea of trying to provide a local spot near most major areas of the Upstate, and after providing a spot close to Pickens and Greenville, Clemson is next. This is a minor addition to the local spots, but one that I used to enjoy when I was an undergrad and grad student at Clemson University...buildering. There are several areas around campus and downtown that offer a few urban climbs. Usually, buildering is frowned upon by onlookers, so try to keep a low profile.
In general, many buildings on campus have a few short problems, and some offer tall aretes and sometimes cracks between adjacent buildings. We used to also climb statues and monuments...when no one was looking:)
There are 2 small areas that were used by several generations of Clemson grads to get a little pump. The first, and most popular, was the incredibly fun sloper wall on the outside of the bottom floor of the "High Rises". It takes a little getting used to, but the traversing and short routes are a fun aside from class or studying. The second, and not as popular, is in the Botanical Gardens. There have traditionally been earthen sculptures in the gardens and many of them had some height. Some of my favorite roof jug routes were right beside the parking area, next to the train. We used to climb up the rock pillars, across the trellis and back down a different pillar. The trellis may not be as strong as it used to be, so test it before you begin the roof sections. Take a walk through the gardens and you will usually find another climb or two that looks interesting...try some routes on the rock bridges or other structures.
A short drive from Clemson are several towers/supports of old bridges that have been bolted and used to climb on by locals. These were sketchy (loose bolts) when I was at Clemson 10 years or so ago, so I would imagine that they have only gotten worse. I'll take a look at them and add it to this post if they are worth a visit.
Nowadays, there is an incredible climbing gym on campus, at Fike, that cant be beat. Check out the link to the Fike Wall on the Upstate Bouldering Links the right. They may not be very big, or they may not be open very often, and they may be crammed full when they are open...but what a great place to get some "real" climbing and bouldering in. With a nice textured Nicros simulated rock wall and normal plywood walls to each side, you get a good variety of climbing. There is usually at least one route in most grades, from 5.7 up to 5.11 or so, and the same with bouldering problems, from V0-V6ish. The routes are changed fairly often, so you get a new challenge every couple of weeks. This is also a place to meet up with other students, faculty and staff that climb and plan some outdoor trips, or sign up for one lead by the Clemson Recreation Department. This is not a public wall, you must be a student, member of the Fike gym, or a guest to use the climbing wall.