Galloping vibrations have been identified as potentially problematic for dry stay cables. Three key issues play a major role in the assessment of the structural damping required to prevent galloping of dry stay cables using a quasi-steady approach: the complex inclined-flow aerodynamics, deviation of the geometry of the high-density polyethylene (HDPE) stay cover with respect to that of a perfectly circular cylinder, and choice of a proper stability criterion. In this paper, the mean aerodynamic force coefficients of a real HDPE plain cable cover measured in a wind tunnel are presented. These were obtained by varying wind speed, yaw angle, and angle of attack, and they represent a complete set of aerodynamic data. Cable irregularities (surface roughness, section distortion, and axis curvature) were quantified and correlated to the measured aerodynamics. The experimental aerodynamic coefficients were used to predict instability using different quasi-steady models from the literature on a reference case. Finally, the different exciting and dissipating mechanisms deriving from the application of one- and two-degree-of-freedom (DOF) stability models, together with the corresponding different response predictions, are discussed in detail. It is shown that cable irregularities and detuning direction exert strong influences on aerodynamic stability and that instability is mainly due to critical Reynolds number effects. Moreover, a comparison of the results shows that the use of multi-DOF models is not justified in this case because one-DOF models prove sufficiently accurate to predict the amount of structural damping required to prevent galloping instability.