What does the no-hair theorem state about the entropy of black holes? So does the no hair theorem say that all classical black holes will have zero entropy? If yes, why?
 A: The no hair theorem is a statement about steady-state vacuum solutions in classical general relativity. It says that once a black hole has settled down to a steady state (by radiating away any dynamic features by gravitational radiation) the solution in the vacuum depends on only a small number of parameters (mass, charge, momentum, angular momentum, and possibly magnetic charge in a theory allowing for magnetic monopoles). In this description the black hole has zero entropy.
The above classical description is believed to be inaccurate however. By bringing in quantum field theory one can mount a strong case that a large amount of entropy is associated with a black hole. One can make this case even without a complete theory of quantum gravity. So we now think that black holes have a non-zero temperature and a very large entropy. The large entropy in turn implies that there is a huge number of microstates which together are consistent with any given set of macroscopic parameters such as $m,Q,{\bf p},{\bf J}$. This implies that by specifying only the macroscopic parameters one does not in fact determine the full details of the state. One could indeed assert that the implication is that instead of 'no hair' a black hole has a huge amount of 'hair', in the sense that a huge amount of information would be needed if one wished to completely specify its state in microscopic detail. Or, to say the same thing another way: if one wished to send a large amount of information from one place to another, one could in principle do so by sending a black hole. This form of communication would only be practical if the sender had the technology to pick one microstate at the sending end, and the receiver had the technology to distinguish among microstates. I mention it as a thought-experiment which serves to clarify the sense in which a black hole may be said to have a lot of 'hair'.
Added remarks in response to comments
Safesphere correctly points out that the no-hair theorem is strictly a theorem about vacuum solutions, so it is questionable whether it applies to any astrophysical object. This is because a black hole formed by gravitational collapse of matter is not quite the same as an 'eternal' black hole. The latter is a solution of the field equations in an artificial case where we ignore cosmic expansion and just come up with a solution which never involved any matter falling in; the horizon is simply there into the infinite past. In classical GR the no-hair theorem applies to artificial solutions of that kind, but one can argue that it does not apply, even in classical GR, to the case of a black hole with matter falling in.
My thought-experiment involving selecting a microstate for a black hole is very much a thought-experiment. It is comparable to having a collection of oscillators whose motions have a variety of phases and amplitudes, and selecting one particular phase and amplitude for each oscillator.
A: The no hair theorem says that a black hole is uniquely identified by few parameters (mass, charge, angular momentum). By itself, there are no implications on the entropy, as different microstates may contribute to the same thermodynamical macrostate described by (M, Q, L).
(Disclaimer following comments: the next paragraph is describing an idealised mathemetical version of a black hole, not a physical one)
Still, in the classical picture (no quantum fields, no evaporation) of a black hole with a pointlike singularity at the center, all the absorbed matter fall into the singularity and leave no trace. Therefore there is only a microstate, $W=1$, equal to the macrostate (M, Q, L=0). The statistical entropy of this black hole is $S \propto log(W) = 0$.
