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The dynamic stability of a viscoelastic plate in a supersonic gas flow and subjected to a stochastically fluctuating axial thrust is performed within the concept of the Lyapunov exponent. The constitutive relation is modeled in an integral form by using the Boltzmann superposition principle. The piston theory as a quasi-first-order approximation is used to represent the aerodynamic loading on the plate. The stochastic averaging method is used and the Khasminskii's technique [Khasminskii, R. A., 'Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic System,' Theory of Probability and Its Application, Vol. 12, No. 1, 1967, pp. 144-147 (English translation)] is employed to obtain the stability boundaries. The influence of the various plate and flow parameters and the random loading spectral densities on the stability are investigated.