What properties should a solvent have? (When purifying a product)

I was just wondering, what are the main, important, properties that a solvent should have? (ideally when purifying a product or in general)

You should select an appropriate solvent such that the the desired substance dissolves fully when it is hot, and be sparingly soluble or insoluble in the cold solvent.

Theoretically, you want a solvent such that the impurity has a higher solubility than that of the desired substance. Of course this is usually impossible because you don't know what this impurity is.
But you don't need to be concerned with that

a) you just assume (at least I've been taught) that the impurities have a higher solubility anyway.

b) even if some of the impurities recrystallise, since the impurities are in such minute quantities, the concentration of recrystalled impurities is going to be even lower, in fact, it's likely that the solubility exceeds the quantity of the impurity because it's so tiny.
You can redo the recystallisation method again and again to reduce the percentage of the impurity even more and more

There's probably plenty other properties...