The classic (and fastest) route is Amphitheater - Saddlerock - Greenman (ASG) from the Gregory Canyon parking lot and ending at the base or top of the summit block. It's about 2.2 miles with 2200' of elevation gain. These trails are clearly shown on this map, where the Gregory Canyon TH is located at the apparent intersection of the Baseline and Flagstaff trails, at the base of Flagstaff Road. George Zack has collected a list of fast times. The trail has changed significantly since the September 2013 flood, including some short re-routes and the addition of a ladder in a badly washed out spot. This route has a Strava segment, but note that Strava can be off by numerous seconds, so it's best to record your actual time in addition to posting your Strava track.
In the old days the standard start was Chautauqua Auditorium, which adds roughly 0.5 mile (relatively flat). Bill Briggs reports that his brother, Roger, ran to the summit of Green from the Auditorium in 35m40s in 1984. Recently, we learned of another fast ascent via a non-standard route by Eric Doub on 5/10/1980 (when Doub was 16?): "I ran Green many times in the 1970s and 80s via this route, and I believe my PR [33m20s] was in spring 1980: W side of Baseline at 6th St. - trail past Amphitheater and Saddle Rock - north ridge direct (later became Habitat Conservation Area, I think) back onto Greenman. On the "north ridge direct" portion, the route involved 3rd class scrambling up and over and down to the Greenman Trail. This route became off limits [requires an off-trail permit for the Habitat Conservation Area] in later years. I think it saved a few minutes."
Zack also reports times for other routes as per these links:
georgezack.blogspot.com/2007/01/fkts-green-mountain-boulder-bear-canyon.html (note that the Bear Canyon and Green-Bear trails have been majorly rerouted and are now MUCH longer than in 2011)
Cordis Hall also reported some "obscure" Green Mtn FKTs: