Kink sites play a pivotal role in the growth and dissolution of materials at the solid-liquid interface. Despite this, little is known about the thermodynamic stability of such sites. Calcium carbonate, in the form of calcite, is one of the most abundant biominerals and a natural means of carbon sequestration in the environment. Here we present a complete determination of the standard free energies for all 16 individual kink sites for the significant case of the calcite (10-14)--water interface using both alchemical and pathway-based simulation techniques. The results reveal the importance of distinguishing between real and ideal ion-binding free energies at surfaces, especially for calcite-water where the interfacial potential can alter values by more than 100 kJ/mol per site. Individual kink site stabilities are found to show variations in excess of 60 kJ/mol, which can help explain observed differences in growth rates between the two distinct steps of calcite.