Understanding the complex relationships between enzyme sequence, folding stability and catalytic activity is crucial for applications in industry and biomedicine. However, current enzyme assay technologies are limited by an inability to simultaneously resolve both stability and activity phenotypes and to couple these to gene sequences at large scale. Here we developed Enzyme Proximity-Seq (EP-Seq), a deep mutational scanning method that leverages peroxidase-mediated radical labeling with single cell fidelity to dissect the effects of thousands of mutations on stability and catalytic activity of oxidoreductase enzymes in a single experiment. We used EP-Seq to analyze how 6,387 missense mutations influence folding stability and catalytic activity in a D-amino acid oxidase (DAOx) from R. gracilis. The resulting datasets demonstrate activity-based constraints that limit folding stability during natural evolution, and identify hotspots distant from the active site as candidates for mutations that improve catalytic activity without sacrificing stability. EP-Seq can be extended to other enzyme classes and provides valuable insights into biophysical principles governing enzyme structure and function.