Knowledge of the dominant singular behavior is crucial to any computer-based stress analysis of finite regions which harbor sharp composite corners. Most numerical methods encounter the problems of lack of convergence and poor accuracy in such regions because of the unbounded stresses (or even unbounded stress derivatives). The introduction of design requirements based on the concepts of linear elastic analyses has made it particularly important that the efficiency of numerical methods be improved when applied to singular problems. Williams treated the plane, elastic infinite wedge for several combinations of homogeneous boundary conditions. The Airy stress function used by Williams is not sufficient whenever the stress singularities are not of the power type, O(r(-lambda)). By appropriately augmenting the class of biharmonic stress functions, Dempsey and Sinclair showed that stress singularities O(r(-lambda) ln r) also may occur for homogeneous boundary conditions. The present paper examines particular cases in which these power-logarithmic singularities occur; the objective is to leave the reader with an intuitive grasp of the typical geometrical configuration and material combination characterized by power-logarithmic singular behavior. The clamped-free wedge is examined first, followed by several bi-material corner problems, including the tri-sector problems formed by open or closed cracks terminating at a bi-material interface. As shown by this study, the power-logarithmic stress singularity typically occurs on transition loci separating regions of real and complex zeros and is by no means unlikely to occur.