H5 influenza is a potential pandemic threat. Previous studies have identified molecular phenotypes of the viral hemagglutinin (HA) protein that contribute to pandemic risk, including cell entry, receptor preference, HA stability, and reduced neutralization by polyclonal sera. Here we use pseudovirus deep mutational scanning to measure how all mutations to a clade 2.3.4.4b H5 HA affect each phenotype. We identify mutations that allow HA to better bind a2-6-linked sialic acids, and show that some viruses already carry mutations that stabilize HA. We also identify recent viral strains with reduced neutralization to sera elicited by candidate vaccine virus. Overall, the systematic nature of deep mutational scanning combined with the safety of pseudoviruses enables comprehensive characterization of mutations to inform surveillance of H5 influenza.