Rayleigh-Ritz method (RRM) is used in this paper in order to investigate the in-plane vibrations of arches with a variable curvature in particular, the frequency parameters and mode shapes. The trial functions are particular solutions of the sixth order differential equation of arch vibrations corresponding to an opening angle equal to one rad. The investigations are made with the follow’s hypotheses: the constant dimensions of the cross-section are small in comparison with the radius, the arch axis is inextensible and the effect of rotary inertia and shear deformation are neglected. The parabolic, catenary, spiral, circular and cycloid arches with different boundary conditions are investigated. The frequency parameters are presented for different types of curved arches with various geometrical properties and boundary conditions and the corresponding mode shapes are plotted. The results are compared with the published literature. The accuracy and relative simplicity of the RRM, applied herein a systematic way with the new choice of trial functions, is established, making it ready to use in more complex situations, such as those of arches with added masses, with non-uniform cross sections or with one or more point supports.