The ongoing pandemic spread of a new human coronavirus, SARS-CoV-2, which is associated with severe pneumonia/disease (COVID-19), has resulted in the generation of tens of thousands of virus genome sequences. The rate of genome generation is unprecedented, yet there is currently no coherent nor accepted scheme for naming the expanding phylogenetic diversity of SARS-CoV-2. Here, we present a rational and dynamic virus nomenclature that uses a phylogenetic framework to identify those lineages that contribute most to active spread. Our system is made tractable by constraining the number and depth of hierarchical lineage labels and by flagging and delabelling virus lineages that become unobserved and hence are probably inactive. By focusing on active virus lineages and those spreading to new locations, this nomenclature will assist in tracking and understanding the patterns and determinants of the global spread of SARS-CoV-2.