The linear and non-linear stability of sheared, relativistic planar jets is studied by means of linear stability analysis and numerical hydrodynamical simulations. Our results extend the previous Kelvin-Hemlholtz stability studies for relativistic, planar jets in the vortex sheet approximation performed by Perucho et al. (2004a, A&A, 427, 415; 2004b, A&A, 427, 431) by including a shear layer between the jet and the external medium and more general perturbations. The models considered span a wide range of Lorentz factors (2.5-20) and internal energies ( $0.08\,c^2{-}60\,c^2$) and are classified into three classes according to the main characteristics of their long-term, non-linear evolution. We observe a clear separation of these three groups in a relativistic Mach-number Lorentz-factor plane. Jets with a low Lorentz factor and small relativistic Mach number are disrupted after saturation. Those with a large Lorentz factor and large relativistic Mach number are the stablest, due to the appearance of short wavelength resonant modes which generate local mixing and heating in the shear layer around a fast, unmixed core, giving a plausible solution for the problem of the long-term stability of relativistic jets. A third group is present between them, including jets with intermediate values of Lorentz factor and relativistic Mach number, which are disrupted by a slow process of mixing favored by an efficient and continuous conversion of kinetic into internal energy. In the long term, all the models develop a distinct transversal structure (shear/transition layers) as a consequence of KH perturbation growth, depending on the class they belong to. The properties of these shear layers are analyzed in connection with the parameters of the original jet models.