The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces.
Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose Laurent series at $p$ looks like $dz/z^n$ for $n > 1$, and which has no other poles on $X$.
The point of this is as a step towards the proof that the Mittag-Leffler problem can be solved for $X$.