A couple of weeks ago someone mentioned Space Elevators, and I said "Oh, it's the end of August, there ought to be results from a Space Elevator challenge. Weren't they doing that annually?" That started me down a path of research and compiling information. It turned out that the International Space Elevator Association was having their annual conference that weekend in Redmond, but without a competition, and the Japan Space Elevator Association had held their competition a week earlier, but there didn't seem to be a place where results were compiled.
As I collected links and results and cross-checked them, and wrote up results, a wikipedia page came into being. I've just posted it, though there's no telling whether it'll last, given wikipedia's unknowable rules and processes.
I've found some details about 9 different competitions, mostly for fast climbers and/or strong tethers. There were a few other competitions that were announced, but which never took place either because no one was ready, no one qualified, or the site fell through.
The most recent climbing results were in Japan in August, where two different teams built climbers that climbed more than a kilometer. The most recent Tether competition I found was in 2011, but I haven't found results for any finisher since 2007, which I think, means the competitors haven't been beating the existing benchmark by a large enough margin. I'm disappointed that the best finishers haven't been published, so we could see how capabilities are growing each year.