  For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the
plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of
any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to
saturated graphs in a graph class as introduced by Tur\'an (1941) and
Erd\H{o}s, Hajnal, and Moon (1964). We focus on $k$-planar drawings, that is,
graphs drawn in the plane where each edge is crossed at most $k$ times, and the
classes $\mathcal{D}$ of all $k$-planar drawings obeying a number of
restrictions, such as having no crossing incident edges, no pair of edges
crossing more than once, or no edge crossing itself. While saturated $k$-planar
drawings are the focus of several prior works, tight bounds on how sparse these
can be are not well understood. We establish a generic framework to determine
the minimum number of edges among all $n$-vertex saturated $k$-planar drawings
in many natural classes. For example, when incident crossings, multicrossings
and selfcrossings are all allowed, the sparsest $n$-vertex saturated $k$-planar
drawings have $\frac{2}{k - (k \bmod 2)} (n-1)$ edges for any $k \geq 4$, while
if all that is forbidden, the sparsest such drawings have
$\frac{2(k+1)}{k(k-1)}(n-1)$ edges for any $k \geq 6$.
