  We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal
algebra $\mathcal E_{m|n}$ associated to Lie superalgebra
$\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative
subalgebras $\mathcal B_{m|n}(s)$ depending on parameters
$s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularity
conditions. We show that $\mathcal B_{m|n}(s)$ is a free polynomial algebra and
give explicit generators which conjecturally correspond to the traces of the
$s$-weighted $R$-matrix computed on the degree zero part of $\mathcal E_{m|n}$
modules of levels $\pm 1$.
