I win. Period. The reach, ease-of-use, ability to be quickly learned, and ability to be easily made makes it the superior weapon for melee warfare.
Between just the rapier and katana, I’d give it to the katana. You can survive a blow from a rapier, most likely, but a katana will end you. I watched a scientific test pitting an English claymore against a katana, and the katana fared better against ice, leather, and metal. It sliced them open easily.
However, the polearm class is more effective, overall.
Axes have their own advantages. They have immense chopping power, and some crushing damage, but they cannot stab, and lack on the defensive front (harder to parry with). Furthermore, they’re somewhat difficult to master, although not very.
Maces can smash through armor and deliver a lot of crushing force, which can cause internal damage without having to break through defenses. They’re heavier, generally, and take some skill to use. They also lack in the parrying department, can’t stab, and take some skill to use.
Swords are more balanced that the previous two. They have chopping and stabbing, and are superior in parrying. While they don’t deliver the sheer crushing/chopping force of their competitors, they can still tear armor up, and cause some serious damage. They do take a lot of skill to master, however.
Hammers are, more or less, similar to maces, overall.
The two-handed variants of these weapons amplify their weaknesses and strengths. The mace is even heavier, has immense crushing power, but still has the pitfalls of its one-handed counterpart. The axe is very similar, as is the sword.
The polearm class, being composed of pikes, halberds, spears, and other weaponry, is another story.
They have a lot more reach than the other weapons. Generally, they are easier to make, deploy in mass, and master. There’s a reason they were the most commonly used weapon in old times (swords were generally only used by knights, nobles, and samurai, due to the usage of resource and skill needed to forge them). You can keep your guard up with a shield while thrusting, if you use the polearm correctly. However, they generally lack in the department of defense, themselves. Try parrying or blocking a sword with a spear, and tell me how it works out.
You can get some easy kills with a spear, if you stab in key points, but you might just be peppering armor with weak hits. That might not matter, if you keep your distance. The spear is generally much longer than a sword, so a knight would have problems hitting a polearm-wielder.
It’s all just a matter of opinion, of course, but I favor the spear in the art of melee combat.
That is, if boxing isn’t allowed.