Slicing is one of the most fundamental geometric operations in Computer-Aided Design and Manufacturing (CAD/CAM). This paper presents a practical slicing algorithm that handles singular cases in a simple and robust manner by making very small movements of vertices. The direction of the shifting is based on the manufacturing methodology of either material removal or deposition as well as singularity types. We evaluated our algorithm by implementing a slicer and applying it to various polyhedral models. The test results demonstrated that the proposed algorithm performed better than conventional slicing algorithms.