We describe the point spectra of some dissipative version of the celebrated ``Rayleigh loaded string'', an elastic string of finite length carrying a number $n\ge1$ of equally spaced, equal point masses, which is a basic model that exhibits a band structure and appears in many applied areas. We consider the case in which the dissipation is due to a viscous damping due to the interaction string-environment, a standard model for internal visco-elastic dissipation (the Kelvin--Voigt model), and their combined presence. We show that the point spectrum of each of these damped versions of the Rayleigh loaded string is a continuous deformation of the point spectrum of the unloaded elastic strings with that damping and that presents a band structure similar to that of the undamped case. We also provide explicit analytical expressions for the eigenfunctions, for any value of $n$.