Magnetohydrodynamic (MHD) surface waves may decay via a process called ''resonance absorption,'' which is a candidate for solar coronal heating. Recently, Hollweg, Yang, Cadez, and Gakovic studied the effects of velocity shear on the rate of resonance absorption of incompressible MHD surface waves. We extend the theory to a compressible but cold plasma, which is the case more applicable to the solar corona. We find that the rate of resonance absorption can be either increased or decreased significantly by velocity shear. We also find that there can exist resonances which lead to instability of the surface mode at values of the velocity shear below the Kelvin-Helmholtz threshold. However, the resonance instability usually occurs when the density ratio across the surface is less than about 0.1. The resonant instability may therefore not be important in the solar corona, though resonance absorption remains an attractive possibility for coronal heating. ¿ American Geophysical Union 1991