There are several definitions of the number of physical degrees of freedom with varying degree of applicability. Once you have a theory in one of the maximally symmetric spaces (Minkowski, de Sitter, anti-de Sitter), it is a good idea to associate particles with unitary irreducible representations (uirrep) of the corresponding symmetry group. Then, the solution space of a given wave equation (modulo gauge transformations, if present) should carry an uirrep. The dimension of what is called Wigner's little group (in flat space) gives the number of degrees of freedom. This not only gives you the quantity, but also the 'quality' and you can distinguish, say, graviton (as massless spin-two) from two spin-zero.
Once you are not in a maximally symmetric spaces a good idea is to count the number of independent (pairs of) functions that determine the Cauchy problem. For example, one can use the Hamiltonian analysis to do that.
Coming to your question. What is usually studied as massless spin-s fields in d-dimensional maximally symmetric spaces (they differ by the value of the cosmological constant). For these type of fields there are no discontinuities related to the value of the cosmological constant and one can apply any of the approaches above. For example, the simplest is go back to flat space, where the Wigner's group is $so(d-2)$ for massless fields and the number of degrees of freedom is given by the dimension of a rank-$s$ tensor of $so(d-2)$. This should be the quickest way. Another way is to look at some standard equations to describe a massless spin-$s$ field, e.g. Fronsdal equations and impose the light-cone gauge to reveal the physical degrees of freedom. In $4d$ the answer is always $2$ degrees of freedom for a spin-$s>0$ field.
Note that Vasiliev theory does not exist beyond free level (as holographic higher spin theories, in general, due to the nonlocality problem).
