Thermodynamic stability and mutational robustness of secondary structure are critical to the function and evolutionary longevity of RNA molecules. We hypothesize that natural and artificial selection for functional molecules favors the formation of structures that are stable to both thermal and mutational perturbation. There is little direct evidence, however, that functional RNA molecules have been selected for their stability. Here we use thermodynamic secondary structure prediction algorithms to compare the thermal and mutational robustness of over 1000 naturally and artificially evolved molecules. Although we find evidence for the evolution of both types of stability in both sets of molecules, the naturally evolved functional RNA molecules were significantly more stable than those selected in vitro, and artificially evolved catalysts (ribozymes) were more stable than artificially evolved binding species (aptamers). The thermostability of RNA molecules bred in the laboratory is probably not constrained by a lack of suitable variation in the sequence pool but, rather, by intrinsic biases in the selection process.