Why even the most volatile solvents leave a residue after evaporating? After a solvent evaporates off of a surface there's always some residue left in the form of a strain.

Why it is the case even for highly volatile solvents such as acetone, toluene etc?
If the surface material is clean and neither soluble nor reactant in any way with the solvent, then it is probably a solvent purity issue. If you have a good micro-balance, you could pour a known volume of the solvent into a pre-weighed light-weight, i.e. aluminum foil, container, dry it, and re-weigh. From that you can calculate the non-volatile solids content of your solvent and compare it to the specifications.