To understand the population genetics of structural variants (SVs), and their effects on phenotypes, we developed an approach to mapping SVs, particularly transpositions, segregating in a sequenced population, and which avoids calling SVs directly. The evidence for a potential SV at a locus is indicated by variation in the counts of short-reads that map anomalously to the locus. These SV traits are treated as quantitative traits and mapped genetically, analogously to a gene expression study. Association between an SV trait at one locus and genotypes at a distant locus indicate the origin and target of a transposition. Using ultra-low-coverage (0.3x) population sequence data from 488 recombinant inbred Arabidopsis genomes, we identified 6,502 segregating SVs. Remarkably, 25% of these were transpositions. Whilst many SVs cannot be delineated precisely, PCR validated 83% of 44 predicted transposition breakpoints. We show that specific SVs may be causative for quantitative trait loci for germination, fungal disease resistance and other phenotypes. Further we show that the phenotypic heritability attributable to sequence anomalies differs from, and in the case of time to germination and bolting, exceeds that due to standard genetic variation. Gene expression within SVs is also more likely to be silenced or dysregulated. This approach is generally applicable to large populations sequenced at low-coverage, and complements the prevalent strategy of SV discovery in fewer individuals sequenced at high coverage.