The University's Pleasance sports centre (often known as the CSE) has bouldering and top-roping facilities open to members of the gym, their guests, University students and staff. You'll nearly always meet yummicks there since it's so close to the main university area and just down the street from GBH. A day pass with gym entry is currently £5, and EUMC members can purchase a discounted pass for a year's climbing.
Further pricing information, times of opening and contact details can be found on their website.
Alien One/Rock (A1)
Housed in a former church building, Alien Rock was Scotland's first dedicated indoor climbing centre. It offers an exciting and interesting range of indoor climbing (top roping, sport, and bouldering!), with roughly 200 routes at a time which are cleaned and changed every four months.
Alien Two (A2)
Alien Rock 2 is the main, larger bouldering facility for Alien Rock with 100s of different problems. It boasts 300sqm of varied angle walls, including an 8m long roof. Plenty to keep both novice and experienced climbers entertained!
Both A1 and A2 can be easily accessed by bike or by bus. For more info on Alien visit their website.
Ratho is enormous -- tons of routes up to 30m high, with top-roping, leading, and bouldering all available. It's popular for endurance training, especially in midwinter -- but bring your belay jackets as it can get really cold. Unfortunately, it isn't well served by public transport (two buses and a walk from central), but it's only a 40-minute cycle along the canal.
For more info visit their website.