The paper considers the problem of the equilibrium stability of the first and second kinds of the nonholonomic dissipative systems with a bilateral and unilateral constraints. We have proved the theorems on the instability under the following the assumptions that: the constraint equations, the kinetic and potential energies and the Rayleigh's dissipation function are infinitely differentiable functions; in the equilibrium position the potential energy has not the minimum. The suggested theorems obtained represent an extension to nonholonomic systems of some results [1], [6], [9], referring to the equilibrium stability of holonomic systems. The proving technique will be similar to that used in the paper [1].