The paper is devoted to the analysis of robotic manipulator behavior under internal and external loadings. The main contributions are in the area of stability analysis of manipulator configurations corresponding to the loaded static equilibrium. In contrast to other works, in addition to usually studied the end-platform behavior with respect to the disturbance forces, the problem of configuration stability for each kinematic chain is considered. The proposed approach extends the classical notion of the stability for the static equilibrium configuration that is completely defined the properties of the Cartesian stiffness matrix only. The advantages and practical significance of the proposed approach are illustrated by several examples that deal with serial kinematic chains and parallel manipulators. It is shown that under the loading the manipulator workspace may include some specific points that are referred to as elastostatic singularities where the chain configurations become unstable.