Structural variants (SVs) are known to play important roles in a wide variety of cancers, but their mutational origins and functional consequences are still poorly understood. The highly nonrandom distributions of these variants across tumour genomes are often assumed to reflect selective processes, but mutation rates can vary by orders of magnitude and often reflect the underlying chromatin structure at a locus. The inference of SVs under selection for enhanced tumourigenesis therefore remains challenging, though identifying such variants may lead to new diagnostic and therapeutic targets. Using experimentally derived mutation data we derive the first quantitative models of double strand break (DSB) frequency across the human genome, based upon underlying chromatin and sequence features. These models provide high predictive accuracy, and models trained in one cell type can be successfully applied to others. We show that most SV ‘hotspots’ (harbouring unusually high SV breakpoint frequencies) seen across a variety of tumour sequencing studies are broadly consistent with DSB model predictions. Using model predictions as a proxy for susceptibility to mutation in tumours, many SV hotspots appear to be adequately explained by selectively neutral mutational bias alone. However, a fraction of hotspots show SV breakpoint frequencies that are unexpectedly high given their predicted susceptibility to mutation, and are therefore credible targets of positive selection in tumours. In contrast, hundreds of regions across the genome show unexpectedly low levels of SVs, given their relatively high susceptibility to mutation. These novel ‘coldspot’ regions appear to be subject to purifying selection in tumours. Both the hotspot and coldspot regions predicted in this manner show intriguing enrichments for genes and regulatory elements.