  We prove real analyticity of all the streamlines, including the free surface,
of a steady stratified flow of water over a flat bed in the absence of
stagnation points, with a H\"older continuous Bernoulli function and a H\"older
continuously differentiable density function. Furthermore, we show that if the
Bernoulli function and the density function possess some Gevrey regularity of
index s, then the stream function admits the same Gevrey regularity throughout
the fluid domain; in particular if the Gevrey index s equals to 1, then we
obtain analyticity of the stream function. The regularity results hold for
three distinct physical regimes: capillary, capillary-gravity, and gravity
water waves.
