  We present a comprehensive theoretical study of vortex lattice formation in
atomic Bose-Einstein condensates confined by a rotating elliptical trap. We
consider rotating solutions of the classical hydrodynamic equations, their
response to perturbations, as well as time-dependent simulations. We
discriminate three distinct, experimentally testable, regimes of instability:
{\em ripple}, {\em interbranch}, and {\em catastrophic}. Under
symmetry-breaking perturbations these instabilities lead to lattice formation
even at zero temperature. While our results are consistent with previous
theoretical and experimental results, they shed new light on lattice formation.
