There's a note on Mt Rainier's trail conditions page
that most of the trails are snow-free:
July 26, 2012: Melt-out has been slowed by cooler than usual tempertaures this spring/early summer. Trails have finally started to melt-out to the point there are only a few areas of concern. However, route-finding is needed in a few places on trails in the park and unseen hazards may be lurking under-foot when traveling on snow. Always check with Park Rangers for trail conditions before heading out into the backcountry.
The Wonderland Trail is 90% snow-free and the Comet Falls Trail
Trails are starting to melt out in the Paradise area. Flowers are blooming as the snow peels back.
is blocked for the foreseeable future.