  We give a complete classification of dynamical invariants in $3d$ and $4d$
Higher Spin Gravity models, with some comments on arbitrary $d$. These include
holographic correlation functions, interaction vertices, on-shell actions,
conserved currents, surface charges, and some others. Surprisingly, there are a
good many conserved $p$-form currents with various $p$. The last fact, being in
tension with `no nontrivial conserved currents in quantum gravity' and similar
statements, gives an indication of hidden integrability of the models. Our
results rely on a systematic computation of Hochschild, cyclic, and
Chevalley--Eilenberg cohomology for the corresponding higher spin algebras. A
new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is
also presented.
