  The purpose of this work is to discuss the well-posedness theory of singular
vortex patches. Our main results are of two types: well-posedness and
ill-posedness. On the well-posedness side, we show that globally $m-$fold
symmetric vortex patches with corners emanating from the origin are globally
well-posed in natural regularity classes as long as $m\geq 3$. In this case,
all of the angles involved solve a \emph{closed} ODE system which dictates the
global-in-time dynamics of the corners and only depends on the initial
locations and sizes of the corners. {Along the way we obtain a global
well-posedness result for a class of symmetric patches with boundary singular
at the origin, which includes logarithmic spirals.} On the ill-posedness side,
we show that \emph{any} other type of corner singularity in a vortex patch
cannot evolve continuously in time except possibly when all corners involved
have precisely the angle $\frac{\pi}{2}$ for all time. Even in the case of
vortex patches with corners of angle $\frac{\pi}{2}$ or with corners which are
only locally $m-$fold symmetric, we prove that they are generically ill-posed.
We expect that in these cases of ill-posedness, the vortex patches actually
cusp immediately in a self-similar way and we derive some asymptotic models
which may be useful in giving a more precise description of the dynamics. In a
companion work, we discuss the long-time behavior of symmetric vortex patches
with corners and use them to construct patches on $\mathbb{R}^2$ with
interesting dynamical behavior such as cusping and spiral formation in infinite
time.
