Singularity of a robot manipulator is one of the obstacles that influences its capabilities. This paper discusses constrained and allowable rotational motion resulting from lost translational freedom when the robot is singular. A convenient method and simple and clear expression to determine the allowable rotational axes and the subspace that they form, under Jacobian singularity, is analyzed and presented. Different configurations, reciprocal screws, and subspaces of allowable-rotational-axes are derived in a case study involving a classic robot. The result is useful in applications involving robot path planning in task space as it extends the usable workspace of rotational axes.