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We report on the strengths and limitations of scanning tunnelling microscopy (STM) when used for characterising atomic-scale features of quasi two-dimensional materials, such as graphene and single layers of hexagonal boron nitride, which may present strong corrugations when grown epitaxially on a substrate with a lattice mismatch. As a paradigmatic test case, we choose single-layer and bilayer graphene on Ru(0001), because their STM images show both a long-range moiré modulation and complex atomic-scale distortions of the graphene lattice. Through high-resolution STM measurements, we first determine with high accuracy the moiré epitaxial relations of the single layer and the bilayer with respect to the metal substrate. In particular, we also provide direct evidence for the existence of AA-stacked bilayer graphene domains on Ru(0001). We then demonstrate that the local strain distribution, as inferred from the same STM images, can be affected by large errors, so that apparent giant strains arise in some regions of the moiré as an imaging artefact. With the aid of density functional theory simulations, we track down the origin of these fictitious distortions in the high directionality of the graphene π -orbital density combined with the large corrugation of the sample. The proposed theoretical model correctly accounts for the observed dependence of the apparent strain on the STM tip–sample separation and on the different degree of curvature of the second graphene layer with respect to the single layer.