Mutational signatures have emerged as powerful biomarkers in cancer patients, with prognostic and therapeutic implications. Wider clinical utility requires access to reproducible algorithms, which allow characterization of mutational signatures in a given tumor sample. Here, we show how mutational signature fitting can be applied to hematological cancer genomes to identify biologically and clinically important mutational processes, highlighting the importance of careful interpretation in light of biological knowledge. Our newly released R package mmsig comes with a dynamic error-suppression procedure that improves specificity in important clinical and biological settings. In particular, mmsig allows accurate detection of mutational signatures with low abundance, such as those introduced by APOBEC cytidine deaminases. This is particularly important in the most recent mutational signature reference (COSMIC v3.1) where each signature is more clearly defined. Our mutational signature fitting algorithm mmsig is a robust tool that can be implemented immediately in the clinic.