= Solution
<Incompressibility> makes each perturbation <velocity potential> a <harmonic function>. A horizontal mode $e^{i(kx+\ell y)+st}$ has $q=\sqrt{k^2+\ell^2}>0$. Decay away from the interface selects $\varphi_1=A_1e^{qz}e^{i(kx+\ell y)+st}$ below and $\varphi_2=A_2e^{-qz}e^{i(kx+\ell y)+st}$ above. The <kinematic boundary condition> gives
$$
qA_1=(s+ikU_1)\zeta_0,\qquad-qA_2=(s+ikU_2)\zeta_0.
$$
Insert these into the <dynamic boundary condition for an inviscid interface> and cancel a nonzero mode amplitude to obtain
$$
\rho_1(s+ikU_1)^2+\rho_2(s+ikU_2)^2+qg(\rho_1-\rho_2)=0.
$$
Writing $\rho_T=\rho_1+\rho_2$ and $\bar U=(\rho_1U_1+\rho_2U_2)/\rho_T$, completion of the square gives
$$
\rho_T(s+ik\bar U)^2-\frac{k^2\rho_1\rho_2}{\rho_T}(U_1-U_2)^2+qg(\rho_1-\rho_2)=0.
$$
Consequently
$$
\boxed{s=-ik\frac{\rho_1U_1+\rho_2U_2}{\rho_1+\rho_2}\ \pm\sqrt{\frac{k^2\rho_1\rho_2(U_1-U_2)^2}{(\rho_1+\rho_2)^2}-\frac{qg(\rho_1-\rho_2)}{\rho_1+\rho_2}}.}
$$
The imaginary advection term does not change growth rates. A mode grows exponentially precisely when the radicand is positive, equivalently
$$
\boxed{k^2\rho_1\rho_2(U_1-U_2)^2>qg(\rho_1^2-\rho_2^2).}
$$
If the upper fluid is heavier, every nonzero wavevector is unstable even without shear: this is <Rayleigh-Taylor instability>. If the lower fluid is heavier, gravity stabilizes sufficiently long waves, but any nonzero velocity difference destabilizes sufficiently short waves with $\ell=0$: this is <Kelvin-Helmholtz instability>. For equal densities, shear destabilizes modes with $k\ne0$. A negative radicand gives purely imaginary frequencies and oscillatory modes; equality is the marginal threshold for exponential growth.
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