Based on the linkage rotatability theory, this article presents the first systematic and unified criteria to identify the existence of branch and dead positions of single closed-loop N-bar planar and spherical linkages or manipulators. It suggests that for any single-loop planar, spherical or even space linkages, the branch problem is irrelevant to the input conditions while the existence of dead positions depends on the type of kinematic chain and also the selection of the input links or joints. The proposed criteria and their simplicity are not affected by the type of applications nor by the use of coupler links as the input. They are also directly applicable to linkages with prism joints. Although the criteria are intended for computer use, the simplicity of these criteria allows one to develop geometric insight for graphic methods.