For $U_1=U_2=0$ $$ c=\pm\left[ \left(\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\right)\frac{g}{k} \right]^{1/2} $$ which is neutrally stable when $\rho_2>\rho_1$. This is the dispersion relation for interface waves in a static medium
Because all wavelengths must be allowed in an instability analysis, we can say that the flow is always unstable to short waves when $U_1 \neq U_2$
The interface becomes a vortex sheet of strength $\gamma=U_2-U_1$. The solution reduces to $$ c=\frac{U_2+U_1}{2} \pm i\left(\frac{U_2-U_1}{2}\right) $$ Here $c$ always has a positive imaginary part for every $k$, so a vortex sheet is unstable to disturbances of any wavelength. The real part $$ c_r=\frac{U_2+U_1}{2} $$ is the phase velocity, equal to the average velocity of the basic flow, consistent with symmetry arguments
