Bonds between molecules in a liquid are also important in determining the extent to which a solid dissolves in the liquid, as illustrated in the following animation:

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For solution to take place, bonds between molecules in the liquid and bonds between particles in the solid have to be broken. New bonds are formed between molecules from the liquid and particles from the solid. Solution is favoured when the bonds that are broken are of similar strength to the bonds that are formed. This idea is sometimes described as 'like dissolves like'.

Are ionic solids such as sodium chloride likely to dissolve more in a polar solvent such as water or a non-polar solvent such as hexane?

Water.

Are non-polar solids such as iodine likely to dissolve more in a polar solvent such as water or a non-polar solvent such as hexane?

Hexane.

Most solids will dissolve to some extent in any liquid, but they are much more soluble in some liquids than others.

The topic of solubility is also covered in the 'Separation and purification' topic of the Analytical chemistry course to which you might also like to subscribe.