  We construct Fock and MacMahon modules for the quantum toroidal superalgebra
$\mathcal{E}_\mathbf{s}$ associated with the Lie superalgebra
$\mathfrak{gl}_{m|n}$ and parity $\mathbf{s}$. The bases of the Fock and
MacMahon modules are labeled by super-analogs of partitions and plane
partitions with various boundary conditions, while the action of generators of
$\mathcal{E}_\mathbf{s}$ is given by Pieri type formulas. We study the
corresponding characters.
