Parallel manipulators might be severely damaged while crossing parallel singularities, because the actuator forces required to keep the platform on a given trajectory might approach infinity. It is thus important to know whether a desired pose of the platform in the workspace can be reached through a singularity-free path or not, and, in the first case, which path should be followed. This paper presents a new numerical procedure for 3UPU and 3UPS spherical wrists, which is able to count and identify the disjoint regions into which the workspace is partitioned by the singularity surface and to assess to which region any two points of the workspace belong. If the two points belong to the same region, a singularity-free path connecting them is found.