  We develop the BRST approach to construct the general off-shell local Lorentz
covariant cubic interaction vertices for irreducible massless and massive
higher spin fields on $d$-dimensional Minkowski space. We consider two
different cases for interacting higher spin fields: with one massive and two
massless; with two massive both with coinciding and with different masses and
one massless fields of spins $s_1, s_2, s_3$. Unlike the previous results on
cubic vertices we extend our earlier result in [arXiv:2105.12030[hep-th]] for
massless fields and employ the complete BRST operator, including the trace
constraints that is used to formulate an irreducible representation with
definite integer spin. We generalize the cubic vertices proposed for reducible
higher spin fields in [arXiv:1205.3131 [hep-th]] in the form of multiplicative
and non-multiplicative BRST-closed constituents and calculate the new
contributions to the vertex, which contain additional terms with a smaller
number space-time derivatives of the fields. We prove that without traceless
conditions for the cubic vertices in [arXiv:1205.3131 [hep-th]] it is
impossible to provide the noncontradictory Lagrangian dynamics and find
explicit traceless solution for these vertices. As the examples, we explicitly
construct the interacting Lagrangian for the massive of spin $s$ field and
massless scalars both with and without auxiliary fields. The interacting models
with different combinations of triples higher spin fields: massive of spin $s$
with massless scalar and vector fields and with two vector fields; massless of
helicity $\lambda$ with massless scalar and massive vector fields; two massive
fields of spins $s, 0$ and massless scalar are also considered.
