There is a well-known argument that if we write the wavefunction as $\psi = A \exp(iS/\hbar)$, where $A$ and $S$ are real, and substitute this into the Schrodinger equation and take the limit $h \to 0$, then we will see that $S$ satisfies the Hamilton-Jacobi equation (for example see http://physics.bu.edu/~rebbi/hamilton_jacobi.pdf).
I understand this, however I feel that I don't understand the claim that this shows that quantum mechanics reduces to classical mechanics in the $\hbar \to 0$ limit. I am confused because I would think that in order to show that QM reduces to CM we would need to show that as $\hbar \to 0$, $|\psi(x,t)|^2$ becomes very narrow and that its center moves in a classical trajectory, ie $|\psi(x,t)|^2=\delta(x-x_\text{classical}(t))$. And it seems that the above argument does not at all show this. In fact, my understanding is that all that matters for the physical measurement of position is $|\psi|^2$ (since this gives the probability distribution) and hence the phase factor $\exp(iS/h)$ seems to not matter at all.
Moreover, some books (see pg 282 of http://www.scribd.com/doc/37204824/Introduction-to-Quantum-Mechanics-Schrodinger-Equation-and-Path-Integral-Tqw-darksiderg#download or pgs 50-52 of Landau and Lifshitz) give a further argument to the one mentioned above. They further say that if $\psi = A \exp(iS/h)$, then $|\psi|^2 = A^2$ satisfies the classical continuity equation for a fluid with a corresponding velocity $dS/dt$, which in the $h \to 0$ limit is equal to the classical velocity.
This argument makes more sense to me. However, I still have some questions about this. (1) I know that there are stationary states whose modulus squared does not evolve in time, which seems to contradict this interpretation of a fluid flowing with velocity v. (2) The fluid interpretation seems to perhaps suggest to me that the wavefunction reduces in the classical limit more to a wave than to a particle. (3) This doesn't show that the wavefunction is narrow.
Best Answer
The subtlety is that an arbitrary wavefunction doesn't reduce to a point of the classical phase space in the limit $\hbar \to 0$ (thinking about phase space makes more sense since in the classical limit one should have definite coordinates and momenta).
So one could ask, which wavefunctions do. And the answer is that the classical limit is built on the so-called coherent states -- the states that minimize the uncertainty relation (though I don't know any mathematical theorem proving that it's always true in the general case, but in all known examples it is indeed so). States close to the coherent ones can be thought of as some "quantum fuzz", corresponding to the quasiclassical corrections of higher orders in $\hbar$.
Example of this for the harmonic oscillator can be found in Landau Lifshits.
Regarding the fluid argument. About your remark (1): the $|\psi|^2$ for the stationary state is indeed stationary, but it still satisfies the continuity equation since the current is zero for such states. Your remarks (2) and (3) are quite right because, as I already said, the classical limit can't be sensibly taken for arbitrary states, it is built from coherent states.
And also I must admit that the given fluid argument indeed doesn't provide any classical-limit manifestation. It's just an illustration that "everything behaves reasonably well" to convince readers that everything is OK and to presumably drive their attention away from the hard and subtle point -- it often happens in $\it{physics}$ books, probably unintentionally :). The problem of a nice classical limit description is actually an open one (though often underestimated), leading to rather deep questions, like the systematic way to obtain the symplectic geometry from the classical limit. In my opinion it is also connected to the problem of quantum reduction (known also as the "wave function collapse").