Gotta get in on this one since I live near Mountain Home and moved there specifically for fly fishing.
There are a number of tailwaters on which to fly fish with White River below Bull Shoals Dam being the largest and most famous with some 100 miles of cold water fishing. The White River has something for everyone with plenty of public access, and great camping at the state park below the dam as mentioned in a previous post. Next is the Norfork River, my homewater and favorite place to fish. It is below Norfork Dam, 4.6 miles long, and was the former world record holder for brown trout at 38 lbs 3 oz (current world record is from Little Red River, below Greers Ferry Dam at Heber Springs, AR at 40 lbs 4 oz.) The Norfork tailwater is somewhat technical, but produces lots of fish, with many browns and rainbows over 20 inches. There are two places for public access: one is below the dam at Quarry Park, a COE campground that is A1 for RVers; many sites have both electric and water, and the other is Ackerman-Handicap Access. Then there is Little Red River which has already been discussed. It's some 25+ miles of cold water fishing, with lots of fish as well. There are several areas available for public access, and JFK Park, a COE campground below Greers Ferry Dam is on the water, has great camping, and some pretty decent fishing. There is also good tailwater fishing below Beaver Dam near Eureka Springs, AR, and Taneycomo below Table Rock Dam at Branson, MO. Gary LaFontaine told me a year or so before he died that Taneycomo was his favorite place to fish for brown trout in the US. The tailwaters hold four species of trout with browns being the largest and most popular, rainbows, cutthroats, and brook trout. There is also some great smallmouth fishing in Crooked Creek and the Buffalo River, both in north central Arkansas, and several rivers and streams in northwest Arkansas.
There is a great book regarding these rivers and fly fishing, Homewaters written by members of the Midsouth South Fly Fishers (I was privileged to author a couple of chapters). Hope this helps. PM me for preferred rod weight, flies, and line to leader to tippet set-up if you're interested.