You need to hit a zombie with each of the following weapons (not necessarily kill them).
This does NOT need to be done in one playthrough (thanks to Misfit119 for confirming).
Royal Flush 1st floor;
Saw Blade: beside air duct as soon as you leave safe room
Box of Nails: first maintenance room after you leave safe room
Metal Baseball Bat: The Man's Sport under cash register
Bow and Arrow: The Chieftain's Hut
Football: Sportrance right wall
Basketball: Sportrance left wall
Lawn Dart: Near sports car underneath dynamite tree (see map)
Royal Flush 2nd floor;
Golf Club: Sportrance left wall beside stairs
Music Discs: Players (shop)
Gems: Eternal Timepieces
Food Court;
Plates: Cucina Donacci restaurant
Palisades Mall First Floor;
Soccer Ball: Kokonutz Sports Town middle of floor
Tennis Racquet: Kokonutz Sports Town right wall
Vinyl Records: Entertainment Isle right wall