From all the research I have done, Katanas do not stand up to advanced Western weapons. The highest quality katanas are good, but the weapon remains fragile enough to be destroyed from something simple as hitting wood. You need to remember that Japan at the time was very poor in metals, especially high-quality metals. Katanas and Samurai weapons were done made to fight unarmored peasants, NOT chop large amounts of bodies. The reason for the stereotype of Katana decapitation is that the blade is less likely to be nicked by a strike to the vulnerable part of the neck. Katana blade making is matched/exceeded by tempered steel, and is completely outmatched by Damascus Steel.
Also, from what I understand, bladed weapons rely more on weight than cutting. If you're looking to cut meat, sharpness is important. However in a large combat setting, you would be constantly chopping bone. Bone is incredibly damaging to sharp weapons, and the lighter Katanas would lose their edge, nick and shatter relatively quickly. A vast majority of Katana tradition revolves around using the weapon as little as possible, and nursing it back to sharpness afterwards. This is the reason why samurai traditionally carried additional swords, such as the Wakizashi, and the Katana was only used in special circumstances. I want to say in honourable duels only, but I'm not certain.
As far as rapiers go, they are MOSTLY superior. The thin, sharp blade is a much better use of tempered/Damascus steel and is more agile, significantly lighter and more versatile. However, as per above, I can't see the weapon dealing too well with bulk body chopping. It's a difference between a duel (where skill is important) and butchering (where heft and repetition are important). When you're fighting hordes of zombies, you're essentially working as a butcher due to the large amount of meat, cartilage and bone you will be cutting apart.
Overall, consider the ineffectiveness of stabbing against zombies. A stabbing weapon aims to inflict pain, cause bleeding, cut muscle and pierce major organs. All of those are traditionally ineffective against zombies. This would also be less effective versus armored enemies (Wolf Spiders) and shapeless enemies (blobs). In all cases, bashing and hacking weapons should prove superior.
The major use of Katanas in warfare has been controlled decapitation, short skirmish and burst duels. Broadsword was the working horse of medieval armies, designed for ongoing combat. Rapiers were more of a specialist weapon, excelling in fighting contemporary armor and unarmored targets.
The better medieval weapons for a zombie apocalypse would be:
- Longsword, due to better construction and greater heft (chopping power, durability)
- Hatchet/Tomahawk for being an excellent chopping wepaon
- Shortsword, due to being versatile, durable and fast
- Morning Star (proper) for being so damn versatile
- Large, spiked maces - I've seen those featured in many places, but the name always varies. Essentially a really large thresher mace.
Here is how I would interpret the existing and mentioned weapons:
- A Katana in unskilled hands would deal moderate damage and shatter quickly when that option is included. A skilled use would significantly decrease speed and increase damage.
- A broadsword would deal heavy damage and be of average speed, but not increase too much with skill
- A Rapier would deal light damage with low skill but be quick. Increased skill would rapidly increase damage.
- A morningstar should be the best weapon against zombies, but have a lower damage ceiling from a rapier
An apocalyptic melee weapon needs to be heavy, fast and not likely to be stuck. It needs to survive continuous hacking against bone and cartilage, and not require excessive maintenance.