I tried to get up to my favorite riding trail in Northern Utah. But the snow has not melted yet. ( May 30th) It will be another couple off weeks until you can really ride that area.
The snow was mostly melted in the sunny areas. But the shadows had snow about a foot deep and int he deep shadows , especially in gullys where the snow had drifted it was belly deep on the horse.
Around my house, I live in the southeastern corner of NM right by TX. The BLM has tons of public lands so that's where I head. I can't really say its trails you pretty much make your own path in the desert. If I feel like trailering I can go to White Sands they have trails there. I have yet to go, but have been told its very nice.