Structural variants (SVs) are a largely unstudied feature of plant genome evolution, despite the fact that SVs contribute substantially to phenotypes. In this study, we discovered structural variants (SVs) across a population sample of 347 high-coverage, resequenced genomes of Asian rice (Oryza sativa) and its wild ancestor (O. rufipogon). In addition to this short-read dataset, we also inferred SVs from whole-genome assemblies and long-read data. Comparisons among datasets revealed different features of genome variability. For example, genome alignment identified a large (∼4.3 Mb) inversion in indica rice varieties relative to japonica varieties, and long-read analyses suggest that ∼9% of genes from the outgroup (O. longistaminata) are hemizygous. We focused, however, on the resequencing sample to investigate the population genomics of SVs. Clustering analyses with SVs recapitulated the rice cultivar groups that were also inferred from SNPs. However, the site-frequency spectrum of each SV type – which included inversions, duplications, deletions, translocations and mobile element insertions (MEIs) – was skewed toward lower frequency variants than synonymous SNPs, suggesting that SVs may be predominantly deleterious. Among TEs, SINE and mariner insertions were found at especially low frequency. We also used SVs to study domestication by contrasting between rice and O. rufipogon. Cultivated genomes contained ∼25% more derived SVs and MEIs than O. rufipogon, indicating that SVs contribute to the cost of domestication in rice. Peaks of SV divergence were enriched for known domestication genes, but we also detected hundreds of genes gained and lost during domestication, some of which were enriched for traits of agronomic interest.