Alright I'll take a stab. Obviously opinions are subject to change as the rest of the game gets made and neither category within the same grade are in any particular order. Rating based on combat usefulness, not on style or non-combat benefits (though some do get mentioned).
Also I'm not factoring in every special trait for every weapon of every category because those can come and go from one progression tier to the next.

S:
-Bow. Did you know that mobs can still hit you while using one of these? Neither did I. Arrows also give it the unique benefit of harnessing multiple elements. If it is ever a bad choice it is because it delivers slow damage for the current enemy in question and nothing else. You could play nearly the whole game safely and comfortably. Just needs another slot or two for ammo.

A:
-Sword. A good all rounder. Slash damage is reliable. The long reaching secondary lunge compliments the standard one handed combo very well. Good for both sweeping in groups and for stabbing from safety.
-One hand clubs. Happens to always have the most effective damage type when the demand for DPS is at its most crucial. The secondary attack is both powerful and practical as you can wind up the golf swing while still closing the gap if you initiate it mid jump, making it much safer to use than it first appears.
-Two hand clubs. Good damage and unmatched crowd control potential. Lacks a shield like the other A-tiers have but who cares if you can open with an area of effect stun?

B:
-Throwing spears. Primary attack is very fast and spammable compared to other one-handers, however pierce damage isn't very hard to come by elsewhere. Throwing your weapon is a nice-to-have utility but unless you save it as a finishing move you're probably just slowing yourself down.
-Polearms. Great range for the rate of attacks but you lose the protection that a shield offers. More pierce damage again. Secondary attack is the coolest attack in the game by far, potentially very effective but also dangerous for the commitment it requires (made a bit safer if started with a running jump). Nice harvesting tool.

C:
-Axes. You may as well look at it the same as a sword but without the secondary. It is capable enough to save you an inventory slot that would have otherwise been reserved for a dedicated melee weapon, but that does nothing for the combat side of things. Although I never leave home without one, I know I could just as easily cover my expedition needs by picking twigs off the ground instead.

D:
-Pickaxes. Slow. Weak. Has pierce damage. Only goes up to iron currently. Has such a niche combat purpose that it is widely considered to be a gimmick. But I think of it as the ultimate slot-savers weapon of choice. For cool, confident but also tedium-loving vikings.

D minus:
-Knives. Just a waste of a slot. Melee stealth attacks are okay for the microscopic fraction of time spent in enclosed spaces or for the occasional fast troll kill but that is it. For most enemies the other weapons already do plenty enough stealth damage. If other weapons can't one-shot from stealth, you usually have time for the second and third hit and if that doesn't kill them it will surely stagger them so big deal. You sacrifice a lot for only a couple of units of carry weight saved. Your general damage is roughly halved and your range sucks. Knives don't have a movement speed penalty but nothing is stopping anyone else from putting away or pulling out any of their larger weapons as needed.