Constant mean curvature planes of finite type in Euclidean 3-space are in correspondence with spectral data, consisting of a hyperelliptic (spectral) curve, two meromorphic differentials, and a line bundle. A class of deformations one can consider are known as Whitham or period preserving deformations. Singularities of Whitham deformations can occur if the differentials have common roots on the spectral curve. In this thesis we are concerned with studying deformations within, and out of, the space of spectral data at which the Whitham equations are singular. We show in a special case that singular Whitham deformations correspond to certain planar graphs on CP1, and study the existence theory of these graphs.