A continuum model is presented that describes the three-dimensional response of an elastic cable that supports a single attached mass. Two asymptotic forms of this model are analyzed for the free, linear response of slack suspensions having small equilibrium curvature (sag) and level supports. The first model, which is valid for relatively small attached masses, assumes that the cable stretches quasi-statically and results in uniform dynamic cable tension. The quasi-static stretching assumption is partially relaxed in the second model which accounts for spatially varying dynamic tension in an approximate manner. The eigen-solutions associated with free response are compared for the two models. Results indicate that the “small mass model” provides excellent approximations to the natural frequency spectrum and vibration mode shapes for most cables and modes of technical interest. A simple criterion is presented which governs the range of validity of the small mass model.