Mutations accumulate in the genome of every cell of the body throughout life, causing cancer and other genetic diseases1-4. Almost all of these mosaic mutations begin as nucleotide mismatches or damage in only one of the two strands of the DNA prior to becoming double-strand mutations if unrepaired or misrepaired5. However, current DNA sequencing technologies cannot resolve these initial single-strand events. Here, we developed a single-molecule, long-read sequencing method that achieves single-molecule fidelity for single-base substitutions when present in either one or both strands of the DNA. It also detects single-strand cytosine deamination events, a common type of DNA damage. We profiled 110 samples from diverse tissues, including from individuals with cancer-predisposition syndromes, and define the first single-strand mismatch and damage signatures. We find correspondences between these single-strand signatures and known double-strand mutational signatures, which resolves the identity of the initiating lesions. Tumors deficient in both mismatch repair and replicative polymerase proofreading show distinct single-strand mismatch patterns compared to samples deficient in only polymerase proofreading. In the mitochondrial genome, our findings support a mutagenic mechanism occurring primarily during replication. Since the double-strand DNA mutations interrogated by prior studies are only the endpoint of the mutation process, our approach to detect the initiating single-strand events at single-molecule resolution will enable new studies of how mutations arise in a variety of contexts, especially in cancer and aging.