  We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers
of order $p$ using higher ramification data. This approach replaces the local
root stack structure of a tame stacky curve, similar to the local structure of
a complex orbifold curve, with a more sensitive structure called an
Artin-Schreier root stack, allowing us to incorporate this ramification data
directly into the stack. As an application, we compute dimensions of
Riemann-Roch spaces for some examples of stacky curves in positive
characteristic and suggest a program for computing spaces of modular forms in
this setting.
