Context: Local reductions of the stellar velocity dispersion in the central regions of galaxies are known as sigma-drops (σ-drops). Knowing the origin of these features can lead to better understanding of inner galactic dynamics. Aims: We present a sample of 20 σ-drop galaxies matched with a control sample of galaxies without σ-drop in order to search for correlations between σ-drops and the properties, primarily morphological, of the nuclear zones and discs of their host galaxies. Methods: We study the dust and Hα distribution at 0.1 arcsec scale, using Hubble Space Telescope imaging, in the circumnuclear zones of the two samples of galaxies, searching for differences and trying to establish a link between the nuclear kinematics and the host morphology. We have also considered the CO and H i emission of the galaxies and their luminosity profiles. Results: We classify the two samples following both morphological parameters and the luminosity profiles. We find a larger fraction of nuclear dust spirals and Hα rings in the σ-drop sample. We also find that the fraction of Seyfert galaxies in the σ-drop sample is bigger than that of LINERs and that the reverse is true for the control sample. Conclusions: Our findings are evidence that a σ-drop is very probably due to inflow-induced star formation in a dynamically cool disc, or in a gas ring, shock focused by an inner Lindblad resonance above a certain critical density level. The same mechanism that feeds the nuclear ring or the nuclear disc is probably reponsible for the higher rate of Seyfert galaxies among the σ-drop hosts. Appendices A-D are only available in electronic form at http://www.aanda.org