I think this is an interesting question. I asked it by myself again and again. I work with refrigerants and oils and measure the miscibility behavior and electrical properties of both too. I thought there must be a limit of the difference between the dielectric constants of (let's say) two liquids. At this limit they will become immiscible. Due to the dielectric properties of the liquids (solids) are temperature dependent the complex forms of miscibility gaps arise.

Now I think the topic is much more complicate. You have to consider solvatation and formation of complexes. These are chemical reactions which form new compounds. The solvated or complexed species cannot be characterized by electric measurements because they are formed in the solution and they are only stable in the solution. On the example you gave above, with water and methanol you can see the problem. Water can build two hydrogen bonds and methanol only one. The properties of the solvated compounds, which wil formed by solution of a solid or liquid, must be different although water and methanol are both polar. What about thinking of an alkan-poly-ol as solvent which can build much more hydrogen bonds, but can be less polar on the other side. I think there is no general rule to fix the problem. I hope to help a little bit.

Best Regards

Steffen.

The dielectric constant epsilon measures the dipole strength of the solvent (epsilon is approximately proportional to C*(ap + p0^2/kT), where C is concentration of molecules, ap is molecular polarizability, p0 is permanent dipole moment, T is temperature). The dielectric permittivity is by far the most important factor when you dissolve ions or polar molecules in a polar dielectric, simply because ion-dipole and dipole-dipole interaction is an order of magnitude larger than any other solute-solvent interaction. In such case you can use Onsager's formula for dipoles or Born formula for ions to estimate solubility - these two are giving you the total solute charge - solvent dipole (Born) or solute dipole - solvent dipole (Onsager) interaction.

But there are other possibilities. For example, your solvent may be a conductor. Then the ion-ion interaction is far more important than the ion-dipole. For this reason, if you consider the solvent ability of ionic liquid or a liquid metal, the most important characteristic is the ionic strength (~C*e^2) or anything related, e.g., conductivity - just epsilon won't be enough for an ionic liquid. You can change the solubility of ionic crystals by two orders of magnitude if you use as a solvent a concentrated electrolyte solution instead of pure water, just through the Debye-Hueckel salting-in effect (ion-ion interaction), and there is the similar Debye salting-in of dipoles (solute dipole-solvent ion interaction).

Further to this, if your solvent is non-polar insulator (e.g., benzene, cyclohexane), i.e. they don't contain neither dipoles nor ions, you should look at their quadrupoles next. Benzene has a massive quadrupole moment, unlike cyclohexane, and although all other properties of theirs are quite similar, benzene is a much better solvent for polar species (due to solute dipole-solvent quadrupole interaction) than cyclohexane. In this case, you need not only the polarity of your solvent, but also its quadrupolar strength ~C*(aq + q0^2/kT), where aq is molecular quadrupolarizability and q0 is permanent quadrupole moment (and I'm so in love with quadrupolarity! - http://scitation.aip.org/content/aip/journal/jcp/144/11/10.1063/1.4943196).

And then, you have a whole bunch of other system-specific interactions - steric, hydrogen bonds, maybe even electron transfer - i.e., chemical interaction between the solvent and the solute.

So, in short, epsilon can be a good measure for the solvent ability for a very large class of couples solvent-solute (polar dielectric as a solvent for polar or ionic solute without specific interactions). Outside this class, you need to consider more bizarre solvent characteristics.

Thanks for help. To be honest, I don't quite understand what you said as a non-chemical guy, Radomir. I will check other materials. 

Steffen, for this question, I would only consider the solubility among non-reacting chemicals. I know reaction will get everything complex. 

Dear Zheng, if you solve a solid in a solvent (e.g. water) everytime a chemical reaction will occur. This is a rule, due to breaking bonds in the solid and forming the solvatation shell of each ion are chemical reactions. Allthoug these processes are reversible the count as chemical reaction. As I said, the electrical properties cannot be measured of the species which you will get in the solutions (e.g. solvatized ions). This is one problem what  Radomir (by the way an excellent answer) and me want to explain. I wish you all the best for your experiments.

Best Regards

Steffen.