Abstract The problem of dynamic stability of viscoelastic plates with any kernel of relaxation is reduced to the investigation of stability of the trivial solutions of a set of ordinary integro-differential equations with periodic coefficients. Using the Laplace integral transform, an integro-differential equation is reduced again to the integro-differential equation of which the main part coincides with the damped Hill equation. Changing this equation for the system of two linear equations of the first-order and using the averaging method, the monodromy matrix of the obtained system is constructed. Considering the absolute value of the eigenvalues of monodromy matrix is greater than unit, the condition for instability of trivial solution is obtained in the three-dimensional space of parameters corresponding to the frequency, viscosity and amplitude of external action. Analysis of form and size of instability domains is carried out.