  We consider singularities in the ElectroHydroDynamic equations. In a regime
where we are allowed to neglect surface tension, and assuming that the free
surface is given by an injective curve and that either the fluid velocity or
the electric field satisfies a certain non-degeneracy condition, we prove that
either the fluid region or the gas region is asymptotically a cusp. Our proofs
depend on a combination of monotonicity formulas and a non-vanishing result by
Caffarelli and Friedman. As a by-product of our analysis we also obtain a
special solution with convex conical air-phase which we believe to be new.
