Saucy, pert, arch are comparable when they mean flippant and bold rather than serious and respectful in one’s manner or attitude.
Saucy is rarely strongly derogatory though it implies some degree of lack of proper respect. Usually it also implies piquancy and levity with a hint of smartness or of amusing effrontery.
Sometimes it is applied also to birds and small animals on similar grounds.
Pert implies a saucy freedom that may suggest presumption or affectation rather than insolence.
In some contexts the word carries additional implications (as of cleverness or sprightliness) found in its other senses.
Arch usually implies roguish or coquettish audacity or mischievous mockery sometimes carried to the point that it seems forced or awkward.