  Using the BRST approach to higher spin field theories we develop a generic
technique for constructing the cubic interaction vertices for N=1
supersymmetric massless higher spin fields on four, six and ten dimensional
flat backgrounds. Such an approach allows formulation of the equations for
cubic vertices including bosonic and fermionic higher spin fields, and the
problem of finding the vertices is reduced to finding the consistent solutions
to these equations. As a realization of this procedure, we present the
particular solutions for the vertices where the fields obey some off-shell
constraints. It is shown that the supersymmetry imposes additional constraints
on the vertices and singles out a particular subclass of the solutions. As a
concrete application of the generic scheme, we consider supersymmetric
Yang-Mills-like systems in four, six and ten dimensions where the higher spin
fields transform under some internal symmetry group, as well as
supergravity-like systems in the same dimensions.
