Somatic mutations drive the development of cancer and may contribute to ageing and other diseases1,2. Despite their importance, the difficulty of detecting mutations that are only present in single cells or small clones has limited our knowledge of somatic mutagenesis to a minority of tissues. Here, to overcome these limitations, we developed nanorate sequencing (NanoSeq), a duplex sequencing protocol with error rates of less than five errors per billion base pairs in single DNA molecules from cell populations. This rate is two orders of magnitude lower than typical somatic mutation loads, enabling the study of somatic mutations in any tissue independently of clonality. We used this single-molecule sensitivity to study somatic mutations in non-dividing cells across several tissues, comparing stem cells to differentiated cells and studying mutagenesis in the absence of cell division. Differentiated cells in blood and colon displayed remarkably similar mutation loads and signatures to their corresponding stem cells, despite mature blood cells having undergone considerably more divisions. We then characterized the mutational landscape of post-mitotic neurons and polyclonal smooth muscle, confirming that neurons accumulate somatic mutations at a constant rate throughout life without cell division, with similar rates to mitotically active tissues. Together, our results suggest that mutational processes that are independent of cell division are important contributors to somatic mutagenesis. We anticipate that the ability to reliably detect mutations in single DNA molecules could transform our understanding of somatic mutagenesis and enable non-invasive studies on large-scale cohorts.