  A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers
each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each
vertex of $H$ at most $t$ times. We consider the existence of shallow hitting
edge sets and the maximum size of shallow edge sets in $r$-uniform hypergraph
$H$ that are regular or have a large minimum degree. Specifically, we show the
following. Every $r$-uniform regular hypergraph has a $t$-shallow hitting edge
set with $t = O(r)$. Every $r$-uniform regular hypergraph with $n$ vertices has
a $t$-shallow edge set of size $\Omega(nt/r^{1+1/t})$. Every $r$-uniform
hypergraph with $n$ vertices and minimum degree $\delta_{r-1}(H) \geq
n/((r-1)t+1)$ has a $t$-shallow hitting edge set. Every $r$-uniform $r$-partite
hypergraph with $n$ vertices in each part and minimum degree $\delta'_{r-1}(H)
\geq n/((r-1)t+1) +1$ has a $t$-shallow hitting edge set. We complement our
results with constructions of $r$-uniform hypergraphs that show that most of
our obtained bounds are best-possible.
