The present paper is based on the contribution by Niihara and his co-workers devoted to indentation testing of ceramic materials, while it provides new observations of peculiarities registered during nanoindentation of sapphire, GaAs and InGaNAs deposited by MBE-technique. Exploiting previous studies of the spherical indentation in sapphire, the present authors recognized different causes that result in the apparently similar pop-in phenomenon for sapphire and GaAs-based semiconductors. The finite element modeling of the quasi-plastic nanoindentation of the ( 1 1 20) plane of sapphire with the elastically deformable tip confirmed that the deformation of sapphire is governed by twinning which causes pop-in phenomenon, as suggested earlier by Niihara et al. The singularities registered for GaAs-based crystals are associated with dislocation movement within {111} slip bands, which is in contrast to the case of sapphire.