  For ideal fluid flow with zero surface tension and gravity, it remains
unknown whether local singularities on the free surface can develop in
well-posed initial value problems with smooth initial data. This is so despite
great advances over the last 25 years in the mathematical analysis of the Euler
equations for water waves. Here we expand our earlier work (Chin. Ann. Math.
Ser. B 40 (2019) 925) and review the mathematical literature and some of the
history concerning Dirichlet's ellipsoids and related hyperboloids associated
with jet formation and "flip-through," "splash singularities," and recent
constructions of singular free surfaces that however violate the Taylor sign
condition for linear well-posedness. We illustrate some of these phenomena with
numerical computations of 2D flow based upon a conformal mapping formulation
(whose derivation is detailed and discussed in an appendix). Additional
numerical evidence strongly suggests that corner singularities may form in an
unstable self-similar way from specially prepared initial data.
