  The main objective of this work is to construct and classify the most general
classical and quantum $\mathcal{N}=1$ $\mathcal{W}_\infty$-algebras generated
by the same spins as the singlet algebra of $N$ fermions and $N$ bosons in the
vector representation of $O(N)$ in the $N\to\infty$ limit. This type of
algebras appears in a recent $\mathcal{N}=1$ version of the minimal model
holography. Our analysis strongly suggests that there is a one parameter family
$\mathcal{W}_\infty[\mu]$ of such algebras at every given central charge.
Relying on this assumption, we identify various truncations of
$\mathcal{W}_\infty[\mu]$ with, on the one hand, (orbifolds of) the
Drinfel'd-Sokolov reductions of the Lie superalgebras $B(n,n)$, $B(n-1,n)$,
$D(n,n)$ and $D(n+1,n)$, and, on the other hand, (orbifolds of) three
$\mathcal{N}=1$ cosets. After a closer inspection we show that these cosets can
be realized as a Drinfel'd-Sokolov reduction of $B(n,n)$, $D(n,n)$ and
$D(n+1,n)$. We then discuss the implications of our findings for the quantum
version of the $\mathcal{N}=1$ minimal model holography.
