A teaching-playback robot manipulator system whereby the user controls the manipulator through a teaching pendant has been used widely in industrial applications. Kinematic singularity issue becomes an important problem in the control of robot with a teaching-playback system. In this paper, we propose and investigate three singularity avoidance methods for a teaching-playback robot manipulator system. Nonredundancy singularity avoidance (NRSA) attempts to reduce both the position and orientation errors of the end-effector with the same priority. Redundancy singularity avoidance (RSA) attempts to reduce the position error of the end-effector with the first priority and reduce the orientation error of the end-effector with the second priority; Both NRSA and RSA are based on a modification of a Jacobian matrix. Point-to-point singularity avoidance (PTPSA) makes the end-effector pass through a singular region based on joint-interpolated control without maintaining the position and orientation of the end-effector. Experimental case studies are developed to investigate the manipulator performance when the end-effector approaches the wrist and shoulder singularity. The maximal end-effector trajectory error and users' feelings are statistically evaluated and analyzed in the experiment. The results of the experiment show the effectiveness and practice of the proposed methods.