A series of alpha-alumina-supported cobalt spinel catalysts of various expositions of the (100), (111), and (110) planes was synthesized and tested in the preferential oxidation of CO (CO-PROX) reaction. Successful control of the polyhedral shape of the nanospinel active phase was achieved via glycerol and/or Zn addition to the impregnation solution. This allowed for the application of such catalysts for resolving the shape-reactivity relationships in a more controlled fashion in comparison to previous studies, where various catalysts of an entirely unlike origin were used. The catalysts were thoroughly characterized by means of X-ray fluorescence, X-ray diffraction, X-ray photoelectron spectroscopy, Raman spectroscopy, scanning transmission electron microscopy, and temperature-programmed reduction techniques. The CO-PROX catalytic performance of the obtained catalysts was examined in the temperature-programmed surface reaction mode. A strong impact of the spinel nanograin morphology on the activity was observed, with the (100) termination found as the most active among all of the exposed low-index (111) and (110) planes. Furthermore, a linear correlation between the abundance of the surface Co3+ cations and the yield of carbon monoxide oxidation reveals their vital relevance as the catalytically CO-PROX active sites. The results are discussed in terms of a thermodynamic diagram of the surface oxygen, water, and oxygen vacancy stabilities on the Co3O4 most abundant planes, providing the rational background for the superior behavior of the (100) termination in the CO-PROX reaction. The obtained results speak in favor of the involvement of the suprafacial oxygen species in the CO oxidation process.