My weapon of choice when facing the undead -- especially when on the run -- is the M1 Carbine. But firearms may not be enough, particularly in close-combat situations such as clearing a building, moving room-to-room. Or, alternatively, after you've run out of ammunition, which is an inevitability in a full-scale zombie apocalypse.
So you'll want to have a melee combat weapon accessible. Max Books of The Zombie Survival Guide is a firm supporter of the popular monk's spade, a polearm of Chinese origin. Here is a demonstration:
This weapon can be used (at either end) for decapitation strikes against zombies while maintaining a distance from their hands or teeth. Although it does not actually kill a zombie, I think that decapitation is the proper goal, as it is far easier than the two other ways to access and destroy a zombie's brain: (1) pierce the eye socket and reach the brain or (2) puncture the skull and crack open the brain directly. A severed zombie head is still a threat, but it is a greatly diminished threat, as it lacks mobility.
My problem with the monk's spade stems from its primary advantage. The length and weight of the weapon will slow and fatigue its user, as well as prevent one from carrying and using a rifle. I prefer to think of a melee weapon as a backup, but the sheer size of the monk's spade prevents that approach.
That's why I prefer the daito, the Japanese longsword. Invented by Miyamoto Musashi, it is a few inches longer than the traditional katana, giving the zombie fighter additional distance from his opponent while making horizontal slashing strikes. At under four feet in length and three pounds in weight, it is imminently portable by a rifleman.
Jeff the Baptist has noted his preference for axes:
The Shaolin spade is a Brooks thing. Like most polearms, it's pretty heavy. If you want something simple and useful, go with an axe or tomahawk. Good stuff and useful for more than skull splitting.
If you want to get fancy, I'm wondering if a dedicated thrusting blade (like a small sword or rapier) would penetrate the eyesocket into the brain and do some good scrambling. They'd make a lighter and more effective sidearm than a lot of the melee weapons I see suggested for the same jobs. Plus they have more reach than a knife or kukri.
I agree that a thrusting blade, such as a rapier, is probably a good choice, but I think that using an axe is asking for trouble. Cracking open a zombie's skull will permanently eliminate it as a threat, but I'd be very concerned that the blade might become lodged in the skull.
What is your preferred anti-zombie melee weapon?