Modern wood cutting axe - preferably a single piece model. You have option of hitting with the sharp edge or if you are worried about axe getting stuck, just hit with blunt the end. Moderately light and you can use it to do other things than just kill zombies. Decent range and can be used in a tighter spaces.
Swords would be pretty bad against zombies since slash damage would not be effective against the undead and they could get stuck and then bend and break. I f you have to go with a sword, keep in mind that all weeaboo crying in the world wont change the fact that western double edged sword design went far past what the east could offer.
I can see bats and shovels being useful - maybe even dual wielding hammers.