Applying perturbation methods, symbolic computation, and generalizing the solution method, higher-order asymptotic solutions are constructed in Lagrangian variables for several models describing 2D standing wave motions in fluids of various configurations. Three main parameters of the fluid configuration, depth, capillarity, and stratification layer, are considered. The frequency-amplitude dependences are obtained and compared with those known in the literature in Eulerian and Lagrangian variables. The comparison shows that the analytical frequency-amplitude dependences are in complete agreement with previous results known in the literature and with the results obtained for other models. A generalization allows us to investigate critical phenomena for standing waves in fluids of various configurations. Namely, special attention is focused on critical values of one parameter, the fluid depth. The frequency-amplitude dependences are analyzed from the point of view of critical values: critical points and critical curves are determined for several models describing standing waves in fluids of various configurations. © 2013 Korean Society for Computational and Applied Mathematics.