The dispersion relations of surface acoustic waves (SAWs) of layered structures can be measured by quantitative acoustic microscopy, laser acoustic methods, and surface Brillouin scattering. Since methods are available to compute SAW dispersion relations as functions of material properties (direct problem), material properties can be derived fitting the computed velocities to the measured ones (inverse problem). The stability and robustness of the inverse problem solution for an isotropic supported thin film is studied with an appropriate sensitivity analysis. The elastic constants that mainly determine each branch of the dispersion relations are pinpointed: the constants that are more reliably determined in each range of film properties are thus identified. Simulations allow one to estimate the level of experimental errors, either in SAW velocities or in film density and thickness, which still allow a meaningful solution of the inverse problem.