Structural variation (SV), involving deletions, duplications, inversions and translocations of DNA segments, is a major source of genetic variability in somatic cells and can dysregulate cancer-related pathways. However, discovering somatic SVs in single cells has been challenging, with copy-number-neutral and complex variants typically escaping detection. Here we describe single-cell tri-channel processing (scTRIP), a computational framework that integrates read depth, template strand and haplotype phase to comprehensively discover SVs in individual cells. We surveyed SV landscapes of 565 single cells, including transformed epithelial cells and patient-derived leukemic samples, to discover abundant SV classes, including inversions, translocations and complex DNA rearrangements. Analysis of the leukemic samples revealed four times more somatic SVs than cytogenetic karyotyping, submicroscopic copy-number alterations, oncogenic copy-neutral rearrangements and a subclonal chromothripsis event. Advancing current methods, single-cell tri-channel processing can directly measure SV mutational processes in individual cells, such as breakage–fusion–bridge cycles, facilitating studies of clonal evolution, genetic mosaicism and SV formation mechanisms, which could improve disease classification for precision medicine.