  We study the local invariants that a meromorphic $k$-differential on a
Riemann surface of genus $g\geq0$ can have. These local invariants are the
orders of zeros and poles, and the $k$-residues at the poles. We show that for
a given pattern of orders of zeroes, there exists, up to a few exceptions, a
primitive $k$-differential having these orders of zero. The same is true for
meromorphic $k$-differentials and in this case, we describe the tuples of
complex numbers that can appear as $k$-residues at their poles. For genus
$g\geq2$, it turns out that every expected tuple appears as $k$-residues. On
the other hand, some expected tuples are not the $k$-residues of a
$k$-differential in some remaining strata. This happens in the quadratic case
in genus $1$ and in genus zero for every $k$. We also give consequences of
these results in algebraic and flat geometry.
