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Given a two‐generated group of prime‐power order, we investigate the singularities of origamis whose deck group acts transitively and is isomorphic to the given group. Geometric and group‐theoretic ideas are used to classify the possible strata, depending on the prime‐power order. We then show that for many interesting known families of two‐generated groups of prime‐power order, including all regular, or powerful ones, or those of maximal class, each group admits only one possible stratum. However, we also construct examples of two‐generated groups of prime‐power order, which do not determine a unique stratum. [ABSTRACT FROM AUTHOR]