It's altitude dependent. This is the time of year when folks start working on busting trails open, like Jenny Creek.
Lower altitude trails like Slaughterhouse Gulch and Rampart Range Road are probably accessible now, but at Rampart they will close the gates if they have not done so already until about Memorial Day. And Dakan's gate might not open until May 1, I can never remember.
In May you will probably be able to do Barbour Fork (if the gate's not locked) and Bill Moore (at least the lower section). Spring Creek maybe as well.
June will see the opening of trails up to about Summit County. We usually hit up Spring Creek on Fathers Day and Argentine the week or so after, though Argentine usually holds enough snow up high to prevent us from reaching the pass or most of McClellan Mountain.
Snowdrifts can hang out on certain trails well into July, or even August. That's why we head to Moab in May, to take advantage of dry ground there while ours thaws out!