An unstable genome is a hallmark of many cancers. It is unclear, however, whether some mutagenic features driving somatic alterations in cancer are encoded in the genome sequence and whether they can operate in a tissue-specific manner. We performed a genome-wide analysis of 663,446 DNA breakpoints associated with somatic copy-number alterations (SCNAs) from 2,792 cancer samples classified into 26 cancer types. Many SCNA breakpoints are spatially clustered in cancer genomes. We observed a significant enrichment for G-quadruplex sequences (G4s) in the vicinity of SCNA breakpoints and established that SCNAs show a strand bias consistent with G4-mediated structural alterations. Notably, abnormal hypomethylation near G4s-rich regions is a common signature for many SCNA breakpoint hotspots. We propose a mechanistic hypothesis that abnormal hypomethylation in genomic regions enriched for G4s acts as a mutagenic factor driving tissue-specific mutational landscapes in cancer.