Hydrolase that can remove conjugated ubiquitin from target proteins such as p53/tp53. Acts as an essential regulator of p53/tp53 stability: in unstressed cells, specifically deubiquitinates p53/tp53 in the cytoplasm, leading to counteracts MDM2 action and stabilize p53/tp53. Following DNA damage, translocates to the nucleus and deubiquitinates p53/tp53, leading to regulate the p53/TP53-dependent DNA damage response. Component of a regulatory loop that controls autophagy and p53/tp53 levels (By similarity). Thiol-dependent hydrolysis of ester, thioester, amide, peptide and isopeptide bonds formed by the C-terminal Gly of ubiquitin (a 76-residue protein attached to proteins as an intracellular targeting signal). Belongs to the peptidase C19 family. USP10 subfamily.