In spherical-mechanism kinematics, instantaneous pole axes play the same role as, in planar-mechanism kinematics, instant centres. Their locations only depend on the mechanism configuration when spherical single-degree-of-freedom mechanisms are considered. Such a property makes them a tool to visualize and/or to analyse the instantaneous kinematics of those mechanisms. This article addresses the singularity analysis of single-degree-of-freedom spherical mechanisms by exploiting the properties of instantaneous pole axes. An exhaustive enumeration of the geometric conditions which occur for all the singularity types is given, and a general analytical method based on this enumeration is proposed for implementing the singularity analysis. The proposed analytical method can be used to generate systems of equations useful either for finding the singularities of a given mechanism or to synthesize mechanisms that have to match specific requirements about the singularities.