Protease which deconjugates SUMO1, SUMO2 and SUMO3 from some substrate proteins. Has isopeptidase but not SUMO-processing activity (By similarity). Desumoylates ZBTB46 (By similarity). Collaborates with UBQLN4 in the export of ubiquitinated proteins from the nucleus to the cytoplasm (PubMed:29666234). Exhibits palmitoyl protein thioesterase (S-depalmitoylation) activity towards synthetic substrates 4- methylumbelliferyl-6-S-palmitoyl-beta-D-glucopyranoside and S- depalmitoylation probe 5 (DPP-5) (PubMed:35427157). {ECO:0000250|UniProtKB:Q9CQT7, ECO:0000269|PubMed:29666234, ECO:0000269|PubMed:35427157}. This is the function of DESI1 (desumoylating isopeptidase 1, ENSG00000100418).