This paper deals with a class of two-dimensional degenerate parabolic equations in a square. The goals are to obtain Carleman estimates and then establish related controllability results. We assume that the degeneracy occurs only on a part of the boundary. We present well-posedness results and then, under some geometrical assumptions, we build suitable weight functions that allow us to deduce global Carleman estimates. As an application, we prove several null controllability and Stackelberg-Nash null controllability results.