Freezing of water droplets placed on the bare and superhydrophobic surfaces of polymer wedges is studied both experimentally and computationally. Two-dimensional numerical calculations of the transient temperature field in a chilled polymer wedge show that the direction of heat flux from the droplet through the thermal contact region with the wedge differs significantly from that from the normal to the wedge surface. A novel approximate computational model is proposed that takes into account the variable area of the water freezing front in the droplet. This model gives a quantitative estimate of the faster freezing of the droplet on the bare surface. The obtained numerical results agree with the laboratory measurements. The velocity of the crystallization front and the droplet deformation including the so-called freezing tip formation are monitored in the experiment. The direction of the freezing cone axis appears to be noticeably different for the cases of bare and superhydrophobic wedge surfaces. This is explained by the fact that the direction of the freezing cone axis is controlled by the local direction of the heat flux. For a hydrophobic wedge surface, the deviation of the freezing tip from the vertical is smaller, because the reduced thermal contact area reduces the influence of the heat flux direction at the wedge surface.