If you are on the Missouri side, it is US 160. If you are on the Arkansas side, it is Hwy 14. Both are nice rides.
I like Mo 90 between Washburn and Noel. It is in the bottom SW corner of Missouri. It is exceptionally twisty and fun. I usually get to it by going North out of Eureka Springs on 23 to MO 86, run that up to Mo 76 near Cassville, then double back to the turn off for Roaring River State Park (Mo Hwy F) and head West to Mo 112, continuing West to Mo 37 at Seligman. Then make the short jump North to 90. I found out about 90 years ago when talking to a local UPS driver at a gas station in Cassville. He was a rider too. I usually run 90 out and back rather than trying to incorporate it into a loop.
In that same area, the run on Mo 76 between Cassville and Cape Fair is also really nice. It slices and twists through the Mark Twain NF. There is a nice little 50s style cafe at the stop sign in Cape Fair. From there you can continue East to Mo 13 and cut South back down into Arkansas or double back on 76 to Mo 39 to head South. I usually run longer routes so I continue East over to US 160 on the East side of Branson, pick up Mo Hwy H and run that North up to Mo 125. Hwy H is paved and a LOT of fun. I run 125 South to the ferry and cross back into Arkansas, then run Ar 14 and other back roads back West to Eureka Springs.
I actually prefer to stay over in Mountain Home. The roads are just as good, if not better, and there is usually less traffic. You can do several great loops up into Southern Missouri and also several in Arkansas. The area South of I-44 between Springfield and Rolla at Hwy 8 is loaded with great roads.
Also, when in Missouri, there are a lot of lettered county roads. They might also be labeled as highways, like Hwy F and Hwy H. These are smaller than your typical two lane, striped and shouldered highways, but the pavement is usually very good and they are often quite scenic and twisty. On rare occasions, you might encounter very short stretches of gravel/dirt, but that is atypical and usually easy to get through.
Of course if you REALLY want to start finding the gems... get a dual sport
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