I did some research on weapons versus armor. The main conclusions:
TL;DR version: For unmodified Fortress Mode, if you want weapons that will work against any enemy, stick with (steel/adamantine) (battle axes/short swords/spears) and non-adamantine war hammers. Bronze or iron armor is sufficient to protect your dwarves from enemy slashing weapons, but steel or adamantine provide better protection against certain other weapons. Nothing can fully protect your soldiers from morningstars, scourges, large daggers, whips, or war hammers.
Testing was done with completely unskilled human subjects in Arena Mode. Note that in cases where weapons could not penetrate armor, they generally ended gaining weapon skill fairly rapidly, and it generally didn't seem to affect the (lack of) penetration.
There are three major categories of weapon attack- broad edged attacks, narrow edged attacks, and blunt attacks. For convenience, I will refer to these as "slashing", "piercing", and "blunt" attacks.
Slashing weapons are almost entirely incapable of penetrating armor of their "level" or below. Any injuries sustained are generally minor bruises (or, more likely, various damage from lucky bites and punches). The levels go Silver<Copper<(Bronze=Iron)<Steel<Adamantine. Thus, a silver battle axe cannot penetrate any armor, and neither can a copper battle axe. Both bronze and iron battle axes will penetrate copper armor, but they bounce off bronze and iron armor. Steel battle axes can penetrare copper, bronze, and iron but bounce off steel and adamantine, and adamantine battle axes can penetrate any armor but adamantine.
Piercing weapons can penetrate armor made of better material, but the penetration ability varies with their contact area. Of course, if the piercing weapon is made of stronger material than the armor, they are guaranteed to penetrate anyway.
Blunt weapons can again penetrate armor. Their penetration ability seems to vary with impact velocity, contact area, and possibly size.
Conclusions about individual weapon types:
-Edged weapons:
All edged attacks with a CONTACT_AREA of 50 or greater seem to be slashing attacks, incapable of penetrating armor of their level or higher. Note that this includes the "stab" attacks of all swords. Steel or adamantine slashing weapons still work just fine in Fortress Mode, as no vanilla enemy of the dwarves has access to steel armor in significant quantities. If dwarves ever start exporting steel armor in quantity during worldgen, stick with adamantine for guaranteed-to-work weaponry.
Spears and pikes seem to be the weapons with the largest CONTACT_AREA (20) that perform as piercing weapons. However, their penetration ability is not necessarily that great. Adamantine spears can pierce all armor but adamantine armor- in other words, exactly the same as adamantine battle axes. Steel spears cannot pierce steel or higher armor. Armor made of iron and bronze is the point where we see a noticeable difference between spears and battle axes, with silver/copper/bronze/iron spears penetrating iron armor about 5% of the time and bronze armor about 20% of the time (one of the few times we see a noticeable performance difference between bronze and iron). Silver and copper spears can penetrate copper armor, again about 20% of the time. Note that "penetration" in this case could mean, say, fracturing bones through the armor without cutting through it- battle axes can't even do that. Spears also seemed to cause major injuries more of the time (compared to battle axes of the same material) if their material was stronger than the armor material, but be warned that this could be more "tearing the muscle!" hits and fewer "the severed part sails off in an arc!" hits.
The next step is weapons with a CONTACT_AREA of 10, in this case morningstars and scourges (both of which also have an IMPACT_VELOCITY of 2000, twice normal, which doesn't seem to hurt). Any non-adamantine morningstar or scourge can penetrate any armor (yes, that includes adamantine). Steel and adamantine armor both reduce the incidence of major injuries slightly, but even adamantine armor lets major hits through about 20% of the time.
Large daggers can also penetrate any armor regardless of material, but their penetration is lessened by the presence of a slashing attack in addition to their (CONTACT_AREA 5) piercing attack. Watch out for those goblin thieves, they have a nasty habit of headshotting your soldiers right when you weren't expecting it.
-Blunt weapons:
Weapon material is almost entirely irrelevant for blunt weapons as long as they're not adamantine. Copper and silver work well since they're much less useful for edged weapons and armor, especially in Fortress Mode where you can't make silver morningstars.
Whips are the best, as is now common knowledge. Whips seem to flat-out ignore armor, with even adamantine armor providing no significant protection. They only cause serious injuries about 33% of the time, but that's not particularly reassuring when a single serious injury can be crippling (and often has fatal follow-up).
Warhammers are second best. Warhammers again deal serious injuries about 33% of the time through armor, dropping to 22% against steel armor and ~15% against adamantine armor.
Weirdly, other blunt weapons are significantly worse. Maces penetrate armor up to bronze and iron about 20% of the time, and can barely penetrate steel or adamantine armor at all. Mauls and flails seem incapable of penetrating any armor at all.