Microbes drive ecosystems under constraints imposed by viruses. However, a lack of virus genome information hinders our ability to answer fundamental, biological questions concerning microbial communities. Here we apply single-virus genomics (SVGs) to assess whether portions of marine viral communities are missed by current techniques. The majority of the here-identified 44 viral single-amplified genomes (vSAGs) are more abundant in global ocean virome data sets than published metagenome-assembled viral genomes or isolates. This indicates that vSAGs likely best represent the dsDNA viral populations dominating the oceans. Species-specific recruitment patterns and virome simulation data suggest that vSAGs are highly microdiverse and that microdiversity hinders the metagenomic assembly, which could explain why their genomes have not been identified before. Altogether, SVGs enable the discovery of some of the likely most abundant and ecologically relevant marine viral species, such as vSAG 37-F6, which were overlooked by other methodologies.