This paper studies the local stability properties of the motion of mechanical systems with holonomic constraints. A set of linear differential-algebraic equations is used to characterize local constrained motion and to determine local stability properties of an equilibrium of the nonlinear differential-algebraic equations. Stability conditions are developed for linear differential-algebraic equations. Examples are shown to demonstrate that the constraint functions play an important role in stability properties of an equilibrium. Conditions are given that guarantee that constraints will not affect local stability properties.