  The minimal model $\mathfrak{osp}(1|2)$ vertex operator superalgebras are the
simple quotients of affine vertex operator superalgebras constructed from the
affine Lie super algebra $\widehat{\mathfrak{osp}}(1|2)$ at certain rational
values of the level $k$. We classify all isomorphism classes of
$\mathbb{Z}_2$-graded simple relaxed highest weight modules over the minimal
model $\mathfrak{osp}(1|2)$ vertex operator superalgebras in both the
Neveu-Schwarz and Ramond sectors. To this end, we combine free field
realisations, screening operators and the theory of symmetric functions in the
Jack basis to compute explicit presentations for the Zhu algebras in both the
Neveu-Schwarz and Ramond sectors. Two different free field realisations are
used depending on the level. For $k<-1$, the free field realisation resembles
the Wakimoto free field realisation of affine $\mathfrak{sl}(2)$ and is
originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1
Heisenberg vertex algebra), one $\beta\gamma$ bosonic ghost system and one $bc$
fermionic ghost system. For $k>-1$, the argument presented here requires the
bosonisation of the $\beta\gamma$ system by embedding it into an indefinite
rank 2 lattice vertex algebra.
