Sumoylation regulates a wide range of essential cellular functions through diverse mechanisms that remain to be fully understood. Using S. cerevisiae, a model organism with a single essential SUMO gene (SMT3), we developed a library of >250 mutant strains with single or multiple amino acid substitutions of surface or core residues in the Smt3 protein. By screening this library using plate-based assays, we have generated a comprehensive structure-function based map of Smt3, revealing essential amino acid residues and residues critical for function under a variety of genotoxic and proteotoxic stress conditions. Functionally important residues mapped to surfaces affecting Smt3 precursor processing and deconjugation from protein substrates, covalent conjugation to protein substrates, and non-covalent interactions with E3 ligases and downstream effector proteins containing SUMO-interacting motifs. Lysine residues potentially involved in formation of polymeric chains were also investigated, revealing critical roles for polymeric chains, but redundancy in specific chain linkages. Collectively, our findings provide important insights into the molecular basis of signaling through sumoylation. Moreover, the library of Smt3 mutants represents a valuable resource for further exploring the functions of sumoylation in cellular stress response and other SUMO-dependent pathways.