When speaking of a problem that has no solution, do the words insolvable, insoluble, and unsolvable have different shades of meaning? How do you decide which to use?
I don't think there's any possibility of a semantic distinction in the context of problems. The only difference worth pointing out is that insoluble is far more common...
This isn't to say that either of the alternatives are "incorrect", though I personally don't like insolvable (it's just a relatively uncommon/dated negated form). And in fact there's some justification for preferring unsolvable simply because insoluble has another well-known sense...
insoluble [of a substance] - that cannot be dissolved [often, in water]
...but since that sense can't possibly apply to problems, there can be no ambiguity here.
A pedant will of course tell you that "insoluble" relates to "solubility", and talking about an "insoluble problem" is... bizarre at best. Alas, this seems to be one of those words that has been co-opted from its original meaning. Still, I wouldn't use it.
I recall seeing once, but can no longer find, that the difference between "insolvable" and "unsolvable" is whether a problem cannot be solved due to insufficient information or resources, or whether it cannot be solved because no solution is possible. I think "insolvable" had the latter meaning, but without being able to find the original source, I'm not sure.
I would think "un" would link to the person applying the word, and that "in" would be applied in universal terms, that would link to all persons. So 'unsolvable' is with respect to my frame of experience, and insolvable with respect to all. So it's "can I, or can I not solve it," or "can it, or can it not be solved?" (But that's guessing--I've been called an idiot). Curious
I think the problem is that "solve" is not the opposite of "dissolve"; a solution may be either a liquid or an answer. The negative form should depend on the meaning of the positive form, so a substance would be insoluble, but a problem would be unsolvable. However, "insoluble" sounds more erudite and is easier to say, making it doubly attractive to most of us.