H5 influenza is considered a potential pandemic threat. Recently, H5 viruses belonging to clade 2.3.4.4b have caused large outbreaks in avian and multiple nonhuman mammalian species. Previous studies have identified molecular phenotypes of the viral hemagglutinin (HA) protein that contribute to pandemic potential in humans, including cell entry, receptor preference, HA stability, and reduced neutralization by polyclonal sera. However, prior experimental work has only measured how these phenotypes are affected by a handful of the >10,000 different possible amino-acid mutations to HA. Here, we use pseudovirus deep mutational scanning to measure how all mutations to a 2.3.4.4b H5 HA affect each phenotype. We identify mutations that allow HA to better bind α2-6-linked sialic acids and show that some viruses already carry mutations that stabilize HA. We also measure how all HA mutations affect neutralization by sera from mice and ferrets vaccinated against or infected with 2.3.4.4b H5 viruses. These antigenic maps enable rapid assessment of when new viral strains have acquired mutations that may create mismatches with candidate vaccine virus, and we show that a mutation present in some recent H5 HAs causes a large antigenic change. Overall, the systematic nature of deep mutational scanning combined with the safety of pseudoviruses enables comprehensive measurements of the phenotypic effects of mutations that can inform real-time interpretation of viral variation observed during surveillance of H5 influenza.