[Luckenberg99]

I have a doubt about solubility: very often it is said that like dissolves like, that is, that polar substances dissolve in polar solvents, and apolar substances dissolve in apolar solvents. But this is not always true: for example, 1-hexanol is polar (its dipole moment is ≠ 0), but it is not soluble in water, while oxalic acid is apolar (being symmetric, its dipole moment dipole is = 0), but soluble in water.
This means that polarity does not imply water solubility.
That said, what do you rely on to roughly predict solubility?

[Borek]

1-hexanol is polar

Hardly, it has a huge, nonpolar, aliphatic group, -OH is just a small part of the molecule.

while oxalic acid is apolar (being symmetric, its dipole moment dipole is = 0)

Polarity is not just a function of a dipole moment, it also depends on the whole molecule shape and properties of its parts. Oxalic acid has two carboxylic groups, both quite polar. Just because their dipole moments cancel out doesn't mean they don't interact strongly with water.

Plus, polarity is mostly a relative concept - it helps us compare molecules, which one is more polar than the other, but it is not a well defined, quantitative concept.

"Similar dissolves similar" is just a rule of thumb, no wonder it fails. But once you get the polarity concept right it is quite OK, we have no better, general ones that could be used as easily.

[Enthalpy]

Oxalic acid is generally represented as trans-, but I expect some rotation capability around the C-C bond. Even if maybe not over 360°, this would give a strong global dipolar moment.

Anyway, I agree with Borek's local moment as the main explanation, especially because water molecules are small.

[Luckenberg99]

1-hexanol is polar

Hardly, it has a huge, nonpolar, aliphatic group, -OH is just a small part of the molecule.

while oxalic acid is apolar (being symmetric, its dipole moment dipole is = 0)

Polarity is not just a function of a dipole moment, it also depends on the whole molecule shape and properties of its parts. Oxalic acid has two carboxylic groups, both quite polar. Just because their dipole moments cancel out doesn't mean they don't interact strongly with water.

Plus, polarity is mostly a relative concept - it helps us compare molecules, which one is more polar than the other, but it is not a well defined, quantitative concept.

"Similar dissolves similar" is just a rule of thumb, no wonder it fails. But once you get the polarity concept right it is quite OK, we have no better, general ones that could be used as easily.

Actually, the definition of polarity is very precise: a molecule is polar if its total dipole moment is different from 0. This means that all asymmetric molecules are polar. However, the fact that a molecule is polar does not mean that it is necessarily soluble in water: for example, the already mentioned 1-hexanol is overall polar, given that it has a total dipole moment ≠ 0, so it will be able to interact with water through hydrogen bonds, permanent dipole-permanent dipole interactions (i.e., the resulting dipole of water aligns with the resulting dipole of alcohol), dipole-induced dipole interactions, and London forces. However, in 1-hexanol structure, there is only one polar group compared to the rest of the molecule which is non-polar, so London interactions 'override' dipole-dipole interactions and hydrogen bonding and the molecule is insoluble.
 
I think a difference needs to be made between polarity and water solubility. A wholly polar substance may still be water insoluble.

Let me know what you think.