  We study the minimal models associated to $\mathfrak{osp}(1 \vert 2)$,
otherwise known as the fractional-level Wess-Zumino-Witten models of
$\mathfrak{osp}(1 \vert 2)$. Since these minimal models are extensions of the
tensor product of certain Virasoro and $\mathfrak{sl}_2$ minimal models, we can
induce the known structures of the representations of the latter models to get
a rather complete understanding of the minimal models of $\mathfrak{osp}(1
\vert 2)$. In particular, we classify the irreducible relaxed highest-weight
modules, determine their characters and compute their Grothendieck fusion
rules. We also discuss conjectures for their (genuine) fusion products and the
projective covers of the irreducibles.
