We show that universal V - b diagrams provide a powerful tool when analyzing the stationary waveguiding properties of the TE waves guided by a thin film bounded by a saturable nonlinear substrate or cladding. For a wide class of non-linearities, the allowed and forbidden regions of these diagrams, for stationary guided propagation to occur, display a universal pattern, the marginal loci separating different allowed regions from the forbidden ones being simple functions of only the asymmetry measure of the waveguide and the saturation value of the nonlinear permitivity. Relevant information for device design purposes is summarized on a few diagrams, so general waveguiding properties can be immediately read-off from them, and threshold power-independent values of the normalized thickness of the waveguide for a particular kind of guided waves to be allowed are obtained. Qualitative information concerning both the guided power and the stability of guided waves is also included in the diagrams.