How does solubility change with polarity?
Solubility is an experimental proposition. There are rules of thumb that suggest polar solutes dissolve best in polar solvents, and that non-polar solutes dissolve best in non-polar solvents, but this is a RULE of THUMB that might not apply in a given scenario.
Possibly the best solvent for use in organic chemistry is ethyl alcohol, a polar solvent that has a good temperature range, and in which, when the SOLVENT is HOT, most organic solutes have some solubility. But again I stress that the solvent you use is governed by experiment.
And sometimes a powerful solvent such as methylene chloride is layered with a weak solvent such as hexanes, and the solvent layers are allowed to slowly to diffuse.