Single Shot kills can be done with weapons that kill in one hit
If you are killing scabs, I suggest the Dirty Harry, most of the time this kills in one hit.
If it is OD, TNTeddy does great against them, and if they are in a horde, you should get a badge in no time.
I usually earn about 3 Single Shot Badges during each of the Overcharge Night Defenses, and there should be loads of OD Around the barricades if you let them bunch up, your only problem should be the Herkers.
Other overpowered weapons aswell include the hairspray bomb at Level 5, this weapon may be expensive but is worth it, it deploys a bomb and when this bomb explodes, it shoots out another three, and they deal massive damage and can kill up to 30 OD With one shot.
EDIT: Credit to neeker75, the One-Handed Dragon Challenge can unlock 4 One-Shot Badges in 2 minutes, if it doesn't just do the challenge again