Our lack of total understanding of the intricacies of how enzymes behave has constrained our ability to robustly engineer substrate specificity. Furthermore, the mechanisms of natural evolution leading to improved or novel substrate specificities are not wholly defined. Here we generate near-comprehensive single-mutation fitness landscapes comprising >96.3% of all possible single nonsynonymous mutations for hydrolysis activity of an amidase expressed in E. coli with three different substrates. For all three selections, we find that the distribution of beneficial mutations can be described as exponential, supporting a current hypothesis for adaptive molecular evolution. Beneficial mutations in one selection have essentially no correlation with fitness for other selections and are dispersed throughout the protein sequence and structure. Our results further demonstrate the dependence of local fitness landscapes on substrate identity and provide an example of globally distributed sequence-specificity determinants for an enzyme.