The effects of ice formation and accretion on external surfaces range from being mildly annoying to potentially life-threatening. Ice-shedding materials, which lower the adhesion strength of ice to its surface, have recently received renewed research attention as a means to circumvent the problem of icing. In this work, we investigate how surface wettability and surface topography influence the ice adhesion strength on three different surfaces: (i) superhydrophobic laser-inscribed square pillars on copper, (ii) stainless steel 316 Dutch-weave meshes, and (iii) multiwalled carbon nanotube-covered steel meshes. The finest stainless steel mesh displayed the best performance with a 93% decrease in ice adhesion relative to polished stainless steel, while the superhydrophobic square pillars exhibited an increase in ice adhesion by up to 67% relative to polished copper. Comparisons of dynamic contact angles revealed little correlation between surface wettability and ice adhesion. On the other hand, by considering the ice formation process and the fracture mechanics at the ice-substrate interface, we found that two competing mechanisms governing ice adhesion strength arise on nonplanar surfaces: (i) mechanical interlocking of the ice within the surface features that enhances adhesion, and (ii) formation of microcracks that act as interfacial stress concentrators, which reduce adhesion. Our analysis provides insight toward new approaches for the design of ice-releasing materials through the use of surface topographies that promote interfacial crack propagation.