The article presents the results of research on nonlinear vibrations of heterogeneous anisotropic shallow shell structures resting on elastic foundations defined by two-parameter model proposed by Pasternak. First, the heterogeneous anisotropic material properties of shells and two-parameter model of elastic foundations are defined. The behavior of heterogeneous anisotropic shallow shell structures is estimated, taking into account nonlinearity of von Karman type for first time. The governing equations are derived taking into account the geometric and physical relationships of heterogeneous or functionally graded anisotropic shell structures and a two-parameter soil model, and then reduced to a nonlinear ordinary differential equation by applying Galerkin method. Depending on the type of sought deflection function, the Airy stress function is found from particular solutions of the inhomogeneous differential equation. The solution of formulated problem is carried out using the semi-inverse method and the frequency-amplitude dependence is obtained for first time. Since double-curved shallow shells can be transformed into spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panel in special cases, expressions for nonlinear frequencies can also be used for these structural elements. The reliability of results is verified by comparing them with numerical-analytical solutions in the literature.