An ascent of Coxcomb from the south involves two separate technical sections plus some ridge-running. To start, find a deeply inset chimney system on the southwest face and start up a 12' cracked block that leads to a 4th class gully. This steepens to an ~80' section of 5.2 chimney. Emerge on the ridge and head east.
A prominent 30' notch is guarded by a 5.6 wide crack to downclimb. 4th class terrain past the notch leads to the summit. Some people rappel into the notch during the ascent and leave the rope to toprope the crack on their return.
Getting down: Many people reverse the route. Either lead the short 5.6 crack out of the notch if you pulled the rap line, or use the rappel rope to toprope out. Back at the 5.2 chimney, some slung blocks can be used to rap the upper part of the chimney (with one rope) to a point where downclimbing becomes straightforward. Alternately, the entire chimney can be downclimbed; Rosebrough notes that this chimney is actually easier to downclimb than to ascend, and, curiously, this is true.
Reportedly there is also a 2-rope rappel station off the north side of the summit. This station is used by parties who are planning to continue on to climb Redcliff to the north.
A light alpine rack, trending toward some bigger hand-sized stuff, could be useful. Take long poot slings and rings to upgrade the slung-boulder rap anchors if necessary.