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 358 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 an outgroup, and long-read analyses suggest that ~9% of genes from this outgroup 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 -- was skewed toward lower frequency variants than synonymous SNPs, suggesting that SVs are predominantly deleterious. The strength of these deleterious effects varied among SV types, with inversions especially deleterious, and across transposable element (TE) families. Among TEs SINE and mariner insertions were especially deleterious, due to stronger selection against their insertions. We also used SVs to study domestication by contrasting between rice and O. rufipogon. Cultivated genomes contained ~25% more derived SVs than O. rufipogon, suggesting these deleterious SVs contribute to the cost of domestication. We also used SVs to study the effects of positive selection on the rice genome. Generally, the search for domestication genes were enriched for known candidates, suggesting some utility for SVs towards this purpose. More importantly, we detected hundreds to thousands of genes gained and lost during domestication, many of which are predicted to contribute to traits of agronomic interest.