A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric function ${}_m F_n$ takes rational values?

The references to the literature where a lot of such examples listed are apreciated very much too.