Most park waters are fishable now!
The Salmonflies have moved up and out and the fish are more tuned-in on Caddis patterns and large attractors. A good number of fish are feeding on these bugs - a PT, frenchie, or hares ear dropper will also fare well. Dark streamer patterns and hoppers like the Mystery Meat, size 4-8, trailed by a stonefly nymph will get you into a good number of fish.
The Firehole, Gibbon and Madison are all fishing great but are seeing a ton of pressure. Your best bets are to get off the beaten path and pack your bear spray. Woolly buggers are a good streamer option. Green nymphs coupled with a Hare's Ear or Rubberlegs should get into some fish. BWO, Beetles, Adam's, and Caddis patterns are a solid dry fly option and have yielded some nice browns.
You can check out the parks Fishing Home page for dates, licensing and fishing related info HERE.