To understand the population genetics of structural variants and their effects on phenotypes, we developed an approach to mapping structural variants that segregate in a population sequenced at low coverage. We avoid calling structural variants directly. Instead, the evidence for a potential structural variant at a locus is indicated by variation in the counts of short-reads that map anomalously to that locus. These structural variant traits are treated as quantitative traits and mapped genetically, analogously to a gene expression study. Association between a structural variant trait at one locus, and genotypes at a distant locus indicate the origin and target of a transposition. Using ultra-low-coverage (0.3×) population sequence data from 488 recombinant inbred Arabidopsis thaliana genomes, we identified 6502 segregating structural variants. Remarkably, 25% of these were transpositions. While many structural variants cannot be delineated precisely, we validated 83% of 44 predicted transposition breakpoints by polymerase chain reaction. We show that specific structural variants may be causative for quantitative trait loci for germination and resistance to infection by the fungus Albugo laibachii, isolate Nc14. Further we show that the phenotypic heritability attributable to read-mapping anomalies differs from, and, in the case of time to germination and bolting, exceeds that due to standard genetic variation. Genes within structural variants are also more likely to be silenced or dysregulated. This approach complements the prevalent strategy of structural variant discovery in fewer individuals sequenced at high coverage. It is generally applicable to large populations sequenced at low-coverage, and is particularly suited to mapping transpositions.