The Kelvin-Helmholtz (KH) instability of a tangential discontinuity between two compressible plasmas in relative motion is investigated, by solving the dispersion equation for two cases. In the first, neutrals are excluded; in the second, collisions between neutrals and ions are introduced in the form of a drag force in the momentum equation. The velocity of neutrals is assumed to be perpendicular to the interface. In both cases the growth rate of the KH instability is obtained as a function of the density jump between the plasmas. Although it has often been remarked that compressibility should, in general, stabilize a plasma, it is found that this ceases to be true when allowance is made for a significant density jump at the interface. Thus, for a large density jump and a large velocity shear, the instability growth rate in a compressible plasma may considerably exceed the growth rate obtained when incompressibility is assumed. Collisions, it is shown, may either stabilize or destabilize a tangential discontinuity, depending on the change in the product of density and collision frequency (pv), as one moves with the neutrals across the interface; when pv decreases, the instability is enhanced (and vice versa).