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
m
-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as
m
≥
3.
m\geq 3.
In this case, all of the angles involved solve a 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 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
π
2
\frac {\pi }{2}
for all time. Even in the case of vortex patches with corners of angle
π
2
\frac {\pi }{2}
or with corners which are only locally
m
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 from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on
R
2
\mathbb {R}^2
with interesting dynamical behavior such as cusping and spiral formation in infinite time.