We present three complementary methods to study stationary nonlinear solutions in one-dimensional nonlinear metal-dielectric structures. Two of them use an approximate treatment of the Kerr-type nonlinear term taking into account only the leading electric-field component, while the third one allows for an exact treatment of the nonlinearity. A direct comparison of the results obtained with all three models is presented and the excellent agreement between them justifies the assumptions that have been used to construct the models. A systematic study of the configurations made of two, three, or four layers that contain a semi-infinite Kerr-type nonlinear dielectric, a metal film, and linear dielectrics is presented. A detailed analysis of properties, type, and number of solutions in these three types of structures is performed. The parameter ranges where plasmon-soliton waves exist are found. Structures with realistic optogeometric parameters where plasmon solitons exist at power levels already used in spatial soliton studies are proposed and studied.