Structural variants (SVs) are a largely unexplored feature of plant genomes. Little is known about the type and size of SVs, their distribution among individuals and, especially, their population dynamics. Understanding these dynamics is critical for understanding both the contributions of SVs to phenotypes and the likelihood of identifying them as causal genetic variants in genome-wide associations. Here, we identify SVs and study their evolutionary genomics in clonally propagated grapevine cultivars and their outcrossing wild progenitors. To catalogue SVs, we assembled the highly heterozygous Chardonnay genome, for which one in seven genes is hemizygous based on SVs. Using an integrative comparison between Chardonnay and Cabernet Sauvignon genomes by whole-genome, long-read and short-read alignment, we extended SV detection to population samples. We found that strong purifying selection acts against SVs but particularly against inversion and translocation events. SVs nonetheless accrue as recessive heterozygotes in clonally propagated lineages. They also define outlier regions of genomic divergence between wild and cultivated grapevines, suggesting roles in domestication. Outlier regions include the sex-determination region and the berry colour locus, where independent large, complex inversions have driven convergent phenotypic evolution.