The potential of the new extended thin-sheet approximation (ETSA) has been investigated by application to a representative range of 2-D problems. The system of governing equations presented by Medvedev & Podladchikov (1999) for 3-D modelling was reduced to two dimensions and tested on problems involving one- and two-layer systems of Newtonian viscous materials. The application of ETSA in each case included (1) setting boundary conditions, (2) completion of equations by evaluation of coefficients, (3) comparison of equations with governing equations of existing thin-sheet approximations, and (4) linear analysis of small perturbations and determination of their dominant wavelengths. It is shown that most previous approaches can be derived by simplification of an extended system under specified boundary conditions. Linear analyses compare well with exact analytical solutions over a wide range of wavelengths for modelling isostatic adjustment, Rayleigh-Taylor instabilities and the development of instabilities due to lateral compression and extension. These problems cannot be described by the previous generation of thin-sheet approximations. Our results suggest that the new extended thin-sheet approximation (ETSA) will be a powerful tool for the realistic modelling of complicated 2- and 3-D geodynamic structures.