Um, wasn't the rapier (or more accurately, the precursor to the rapier) developed to specifically counteract plate armor?
From what I remember of my history, as soon as plate armor became popular in Europe, sword blades started getting narrower so as to slide between the overlapping pieces of metal.
Nope. They are too light for battlefield use (not to mention, on a battlefield it's a much more brutal affair, no real time for precision shots like that). They were designed for civilians and duels and are often referred to as a type of "dress sword." That is, a courtly sword, or one worn for show and self-defense if needed, but not war.
Attempting to slide a blade between overlapping pieces of metal in movement while that opponent is actively attempting to kill you is, at least to me, absurd. Tackling somebody and attempting to slide a dagger through a spot, maybe. Well-made plate armor fits together with precision, leaving little in the way to slide a blade through. But against a moving target, none at all. You either have to thrust or chop through the armor with a heavy enough weapon (and leverage, such as using two-hands) or batter them.
Weapon design against plate armor was essentially one of power - larger swords (enabling two-handed use), particularly used while thrusting. Longswords were designed to be used primarily two-handed to gain enough power. Of course, other weapons other than swords were the heavy polearms, as the leverage of the long shaft made chopping through the armor feasible. Of course maces and other bludgeoning weapons were another approach, although against full plate, or even a breastplate with mail and a gambeson under either did a decent job of spreading the blow. Other specialty weapons like the estoc (a type of short sword stiff thrusting blade) were also developed. The narrow blade of the estoc serves the same purpose as the bodkin arrowhead. The narrower the blade, the better it was as punching through plate armor. But if you've watched any of the tests of arrows against plate armor that numerous people have tried, a breastplate over mail and a padded gambeson was virtually invulnerable to arrows. Maybe a crossbow, since it could have a much heavier draw weight. But the big advantage of the crossbow was that it required much less training than a bow, thus less skill, but also one of strength. It's very difficult to draw a 100#+ war bow.
Rapiers are designed for quick and nimble attacks against (largely) unarmored opponents, and were developed relatively late.
There is an intermediate type of sword, that are now usually referred to as side-swords, that slowly replaced heavier arming swords, and were essentially military precursors/versions of the civilian rapier.
Here's a decent site on types of swords:
http://www.thearma.org/terms4.htm#.WQap91LMxE4
Keep in mind that for the most part people didn't wander around wearing heavy armor. Plate armor was for decoration, station, and war. In town (or cities) people didn't typically wear armor. Courtesans and nobility might carry a sword, but they became smaller over time, the rapier is one example, which led to the shorter "small-sword" over time.
Town guards (at walls) would be armored in mail, or mail with a breastplate and additional protection for the legs and arms, or perhaps munition plate.
Even in a fantasy world, within a city I would expect that heavy armor is uncommon amongst people in the streets. Most would be in leather, or something like brigandine or a jack of plates. Mail or heavier armor is heavy, hot, and not terribly comfortable to wear all the time. Plate is worse. Not like it's often represented, but for the most part it's worn when people expect to need it. Like war. Or guard duty.
The laws will also have a lot to say about who can draw a weapon, and when. This is less restrictive in a fantasy world than a feudal medieval society, but they will be there. Frontier law (like the wild west) is reasonable in distant towns, but in a large city drawing steel would most likely land you in jail. The idea of duels being more common, on the other hand, make a lot of sense. These would be much safer for the rest of the citizens, and relieves the law from having to deal with personal disputes as well.
Having said all that, the history of weapons and such still has a lot of unknowns and gray areas. Some weapons work better against certain types of armor than others. If you have a different view of how a specific weapon works, that's fine. Go with what works for you. But, people wore (wear) armor because it's effective. IN the case of mail armor or heavier, it was very effective. I like to take that into account. I also like to take into account that certain tactics and resources make a difference.