For a class of nonspherical-wrist manipulators, i.e., redundant or nonredundant, we propose unified singularity analysis and computation-effective avoidance methods. First, we construct a unified model to describe this class of manipulators and derive the kinematics equation and its modified form. Second, the Jacobian matrix of an $n$-degree-of-freedom (DOF) manipulator is partitioned into block triangle form, and the singularity conditions are isolated and collected in a $3\times(n-3)$ submatrix. By analyzing the rank degeneracy conditions of the submatrix, singularity configurations are identified. Third, based on the partitioned Jacobian matrix, the kinematics equation is decomposed into two smaller dimension subequations, only one of which (called singular subequation, as determined by the $3\times(n-3)$ submatrix) contains singularities. Finally, reduced-order approaches and the singularity parameter optimization (SPO) method are presented to plan the singularity-free trajectories by handling the singular subequation. Theoretical analysis shows that the computation costs of the reduced-order approaches are only 1/3–1/2 of the traditional methods. The SPO method is even more efficient than the reduced-order methods. Simulation results verify the proposed methods.