This work presents a method for identifying all the kinematic designs of spatial positioning manipulators that are optimally fault tolerant in a local sense. We use a common definition of fault tolerance, i.e., the post-failure Jacobian possesses the largest possible minimum singular value over all possible single locked-joint failures. The large family of physical manipulators that can achieve this optimally failure tolerant configuration is then parameterized and categorized. We develop a general computational technique to evaluate the resulting manipulators in terms of their global kinematic properties, with an emphasis on failure tolerance. Several manipulators with a range of desirable kinematic properties are presented and analyzed, with a specific example of optimizing over a given class of manipulators that possess a specified kinematic constraint.