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 Dirichlets 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.
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