  We study interactions of higher-spin massless fields $\varphi$ with conserved
currents multilinear in the off-shell matter fields $\phi$. Specifically, we
focus on the 3d case where a slight modification of the $\sigma_-$-cohomology
technique developed earlier is directly applicable to control nontriviality of
the interaction vertices at the convention that the vertices removable by a
local field redefinition of a higher-spin field or having schematic form
$F(\varphi) G(\phi)$, where $F(\varphi)$ is a gauge invariant field strength of
a free higher-spin field, are called deformationally trivial. It is
demonstrated how the $\sigma_-$-cohomology approach can be applied to the
analysis of nonlinear vertices. Generally, deformationally trivial vertices are
not $\sigma_-$-closed while the deformationally non-trivial ones must be in
$H(\sigma_-)$. It is shown that, at least in the $3d$ case, the relevant
cohomology group $H^1(\sigma_-)=0$ and, hence, no deformationally non-trivial
off-shell vertices exist. On the other hand, there exists an infinite class of
deformationally trivial vertices, that includes the vertex recently proposed
for spin three. Our analysis goes beyond the higher-spin vertices allowing to
show that, at least in three dimensions, nonlinear combinations of the
off-shell scalar fields and their derivatives cannot obey non-trivial
equations.
