In dimensions 2 and 3 we prove that the spectral stability bound does not exceed some quantity of the dynamical systems origin. In dimension 2 this implies the spectral bound is equal to the growth bound in $L^2$ of infinitesimal perturbations in an ideal incompressible fluid (joint work with Yuri Latushkin). We discuss the question if linear instability in dimension 2 implies nonlinear (Lyapunov) instability of a smooth Euler equilibrium (joint work with Susan Friedlander).