It seems to me that spin/polarization entanglement is the only spooky one. That's because the entanglement of other observables would happen even in regular classical physics if we made a "wavefunction" consisting of a probability distribution over an unknown variable.
Take position. Start with two classical charges $a$ and $b$ with known positions and unknown velocities at time $t_1$, and let them evolve to time $t_2$. Assume every initial velocity configuration is equally likely.
Now, construct the probability distribution $P(\{\vec r_a, \vec r_b\})$ over final position configurations. I think the math would be pretty hard, but I'm also pretty sure that it would not be separable, that is,
$$P(\{\vec r_a, \vec r_b\}) \ne P_a(\vec r_a)P_b(\vec r_b)$$
That's because, once you know where particle $a$ ended up, you then know (in principle) what the possible two-particle trajectories are that share that value of $\vec r_a$. And you can "count them up" (really more of a path integral I guess) according to their value of $\vec r_b$ and thus come up with $P(\vec r_b | \vec r_a)$, the distribution of $\vec r_b$ given the known value of $\vec r_a$. But if you suppose $a$ ends up somewhere else, call it $\vec r_a'$, then $P(\vec r_b | \vec r_a')$ almost certainly would not be the same.
Hence, the classical version of entanglement. And it results from interaction between the charges, just as in the quantum case. Of course, the fact that the distribution, and thus the entanglement, evolves exactly according to the laws of quantum field theory, is still mysterious, but that's quite apart from the issue of the existence of entanglement. And it at least has the hope of being explained by a local realist theory like the pilot wave.
The exception is spin (and maybe photon polarization?) But that is -- at least partially -- related to the fact that the measurement of spin is inherently quantum. If spin followed the classical tradition, you would get a continuous range of observed values, and the value for the first particle would almost entirely pin down the value you'd get for the second, regardless of the measurement axes used. Hence, they would still be entangled (non-separable probability distributions), but it would be a trivial kind of entanglement that's obviously compatible with local realism.
Now, the quantum nature of one-particle spin measurement could still in principle be described by a classical theory. There would just have to be some mechanism that would align the particle's spin axis with the axis of the detector -- but in order to remain local realist, the change in the particle's angular momentum would have to be somehow absorbed by the detector, or something in the vicinity. So the spookiness enters the picture when we realize that the probability distribution of $b$'s spin correlates with the axis chosen for $a$'s detector, with which $b$ can have no direct local interaction -- thus leading us to conclude that the change in $a$'s angular momentum was transferred to $b$ non-locally.
But there is apparently nothing inherently quantum about measuring position or momentum or energy -- that is, if we imagine a classical underlying theory, we don't have to believe that the nature of the detector changes the value of the classical variable during the measurement process.
And if there is no sudden change of the variable, there is no information that needs to be instantaneously transmitted to the other particle in order to enforce the correlation. Rather, the correlation can be generated by the continuous interaction between the particles via speed-of-light propagation of fields throughout their flight, even though the nature of the fields/waves involved would undoubtedly differ from prior classical theories.
Therefore, why do we have to believe that the entanglement of these non-spin variables is incompatible with local realism?
And if the answer is something to do with the technical details of Bell's theorem, I'd really appreciate if you could try to explain the ideas involved, hopefully enough so I can actually understand how the correlation of e.g. position would violate the inequalities, as this appears to me to be waaay more complicated than spin!
