I'd take an axe over a katana. On one hand, the sword is a hell of a lot less likely to make you waste seconds getting it unstuck from something's neckbone, because really, it evolved over the course of centuries to be really good at not getting stuck in people you're cutting up.
On the other, an axe (or hatchet) is less unwieldy, works better in narrow spaces, doesn't need maintainence or a razor sharp edge, requires less training and precision, and the important bit: It's useful for more than one thing. You will not be using a katana to break down doors or chop firewood. An axe is an indispensable tool with multiple uses in addition to being a handy weapon, and you'll need to be carrying one anyway. More than a little important when you'll want to minimize the amount of weight you're lugging around. See also: crowbar, entrenching tool.
Also, grenades generally kill or disable people by perforating their organs with shrapnel. If they're not using those organs it's kind of not a big deal. You'll have a better chance of hitting something's brain with damn near anything else, and a hell of a lot less likely to injure yourself and others. Same thing with land mines, which mostly mangle legs and/or genitals, forcing an enemy to waste time and resources evacuating the injured (also handy in that sending home legless, dickless kids tends to erode popular support for a war.) Zombies generally don't give a shit about any of that. Fuck explosives.