The singularities of the differential kinematic map, i.e., of the manipulator Jacobian, are considered. We first examine the notion of a "generic" kinematic map, whose singularities form smooth manifolds of prescribed dimension in the joint space of the manipulator. For 3-joint robots, an equivalent condition for genericity using determinants is derived. The condition lends itself to symbolic computation and is sufficient for the study of decoupled manipulators, i.e., manipulators which can be separated into a 3-joint translating part and a 3-joint orienting part. The results are illustrated by analyzing the singularities of two classes of 3-joint positioning robots.