The stability/instability of traveling periodic Stokes waves—the first global-in-time solutions ever discovered for nonlinear quasi-linear dispersive PDEs—is a central, long-standing question in fluid mechanics. In 1967, Benjamin and Feir proposed a famous heuristic mechanism suggesting instability under longitudinal long-wave perturbations, complemented around 1980 by McLean's numerical identification of additional transverse instability. This mini-course will present a mathematically comphensive description of the Fourier-Bloch-Floquet spectral bands for the linearized water wave operator at small-amplitude Stokes waves under 3d longitudinal and transverse wave perturbations. We achieve this by exploiting the Hamiltonian and reversible strucure of the water waves equations, and rigorously combining spectral and perturbation theory, jointly with dynamical systems techniques.