Regional brain morphology has a complex genetic architecture, consisting of many common polymorphisms with small individual effects. This has proven challenging for genome-wide association studies (GWAS). Due to the distributed nature of genetic signal across brain regions, multivariate analysis of regional measures may enhance discovery of genetic variants. Current multivariate approaches to GWAS are ill-suited for complex, large-scale data of this kind. Here, we introduce the Multivariate Omnibus Statistical Test (MOSTest), with an efficient computational design enabling rapid and reliable inference, and apply it to 171 regional brain morphology measures from 26,502 UK Biobank participants. At the conventional genome-wide significance threshold of α = 5 × 10-8, MOSTest identifies 347 genomic loci associated with regional brain morphology, more than any previous study, improving upon the discovery of established GWAS approaches more than threefold. Our findings implicate more than 5% of all protein-coding genes and provide evidence for gene sets involved in neuron development and differentiation.