A quick dissertation on some of my favorite one-handed weapons, their pros and cons, and how to use them effectively. I generally don't use two-handed weapons, as I consider that an open invitation for some unnoticed archer to put a lucky crossbow bolt through my lung.
The Battle Axe
No dwarf should be without one! More seriously, while a battle axe isn't strongly superior in any one category, it's pretty good in several. It has one of the best slicing penetration depths for a single handed weapon, so if your goal is to leave lots of dismembered hands, arms, legs, and heads in your wake, this is a solid option. It can have a hard time with armor however, so harder materials are important: Steel>Iron>Bronze>Copper
Battle axes work best against human or smaller opponents, where it has an excellent chance of dismembering.
Best opening targets are usually lower arms or legs, as the battle axes' cutting penetration makes a broken bone or lost limb very likely on a good hit. It's particularly fun to target the head when you get the opportunity, as nothing displays your skill like your opponent's head arcing through the air.
If you are having trouble with armor penetration or thick fat layers, consider pommel bashing in a pinch, it can actually be quite effective if you're not getting blade penetration.
The Spear
One of the best one-handed weapons against Giant creatures. Spears combine good armor penetration with excellent wound depth, allowing you to inflict mortal wounds against even the largest of beasts. Excellent anti-dragon/hydra/cyclops weapon.
Hardness is far more important than weight for spears, so again: Steel>Iron>Bronze>Copper
The spear is not quite as good for crippling arms and legs, so your best opening target is generally the Lower Torso with all those vulnerable internal organs - a good hit here will often result in your opponent being gutted, which will cause them disabling pain and nausea from which they cannot recover. Upper Torso strikes are good for causing massive bleeding or lung damage, which will further weaken the beast. Head shots are of course likely to be fatal on a direct hit. Extra style points for nailing a cyclops in the eye.
The Morning star
One of the stronger one-handed bashing weapons. The Morning star is theoretically effective against opponents with non-rigid armor (esp. chain mail). Like most bashing weapons, the morning star is good for breaking bones and bruising organs, but it generally will not dismember or cause much bleeding, so while they are good for disabling opponents, it can be harder to finish them off.
For blunt weapons of this sort, both weight and/or hardness is important so: Steel~Silver>Bronze>Iron>Copper
Against human-sized opponents, a strike against the lower limbs or extremities will often crush bones, disabling them. Upper torso shots can break ribs which may tear the lungs. Lower torso shots tend to be less effective, though a direct hit can result in retching or other organ damage. Only a head shot is likely to kill an opponent quickly.
The Whip
Whips are unusual weapons with an insanely small contact area, and very little penetration depth. This makes them very likely to penetrate armor, but the wounds they cause are often superficial - but painful.
Non-intuitively, this makes whips quite effective against armored opponents, in that they can land several painful and injurious early hits in order to reduce their combat effectiveness. As such, they make quite effective 'opening' weapons, for softening up tougher enemies. Opening shots to the lower torso or lower extremities are often effective to soften up an opponent.
The whip is not that effective against enemies that are highly injury resistant however, such as hydras, or creatures with a lot of fat.
I'm not clear on what the best whip material is, but Iron, Steel or Silver are generally effective.
Actually killing an opponent with a whip can take dozens of rounds - and can generally only be accomplished via a direct head-shot, so it's often better to switch to another deadlier weapon once you've taken some of the fight out of them.
Whips are not particularly effective when fighting a horde of goblins, as you risk being overwhelmed before you manage to finish any of them off. On the plus side, unlike most of the weapons above, whips will never get stuck in your opponent, so a very skilled lasher can disable large number of opponents in a short time during a running fight.
Be extremely wary of enemies wielding whips, as these weapons are the most likely to get a lucky shot through your fancy armor and injure you.