  We study analytic properties of the dispersion relations in classical
hydrodynamics by treating them as Puiseux series in complex momentum. The radii
of convergence of the series are determined by the critical points of the
associated complex spectral curves. For theories that admit a dual
gravitational description through holography, the critical points correspond to
level-crossings in the quasinormal spectrum of the dual black hole. We
illustrate these methods in ${\cal N}=4$ supersymmetric Yang-Mills theory in
3+1 dimensions, in a holographic model with broken translation symmetry in 2+1
dimensions, and in conformal field theory in 1+1 dimensions. We comment on the
pole-skipping phenomenon in thermal correlation functions, and show that it is
not specific to energy density correlations.
