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Characterization of somatic mutations at single-cell resolution is essential to study cancer evolution, clonal mosaicism and cell plasticity. Here, we describe SComatic, an algorithm designed for the detection of somatic mutations in single-cell transcriptomic and ATAC-seq (assay for transposase-accessible chromatin sequence) data sets directly without requiring matched bulk or single-cell DNA sequencing data. SComatic distinguishes somatic mutations from polymorphisms, RNA-editing events and artefacts using filters and statistical tests parameterized on non-neoplastic samples. Using >2.6 million single cells from 688 single-cell RNA-seq (scRNA-seq) and single-cell ATAC-seq (scATAC-seq) data sets spanning cancer and non-neoplastic samples, we show that SComatic detects mutations in single cells accurately, even in differentiated cells from polyclonal tissues that are not amenable to mutation detection using existing methods. Validated against matched genome sequencing and scRNA-seq data, SComatic achieves F1 scores between 0.6 and 0.7 across diverse data sets, in comparison to 0.2-0.4 for the second-best performing method. In summary, SComatic permits de novo mutational signature analysis, and the study of clonal heterogeneity and mutational burdens at single-cell resolution.