K-differentials with prescribed singularities

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\ge 0$ can have for $k\ge 3$. These local invariants include the orders of zeros and poles, as well as the $k$-residues at the poles. We show that for a given pattern of orders of zeros, there exists, with a few exceptions, a primitive holomorphic $k$-differential having zeros of these orders. In the meromorphic case, for genus $g\ge 1$, every expected tuple appears as a configuration of $k$-residues. On the other hand, for certain strata in genus zero, finitely many tuples (up to simultaneous scaling) do not occur as configurations of $k$-residues for a $k$-differential.