  In this paper, we study minimal (with respect to inclusion) zero forcing
sets. We first investigate when a graph can have polynomially or exponentially
many distinct minimal zero forcing sets. We also study the maximum size of a
minimal zero forcing set $\overline{\operatorname{Z}}(G)$, and relate it to the
zero forcing number $\operatorname{Z}(G)$. Surprisingly, we show that the
equality $\overline{\operatorname{Z}}(G)=\operatorname{Z}(G)$ is preserved by
deleting a universal vertex, but not by adding a universal vertex. We also
characterize graphs with extreme values of $\overline{\operatorname{Z}}(G)$ and
explore the gap between $\overline{\operatorname{Z}}(G)$ and
$\operatorname{Z}(G)$.
