For culling a bird you won't be eating, dislocation of the neck is a good way to go about it...no blood, no mess, very quick. There are various ways to dislocate a bird's neck...broomstick, by hand(not wringing, as it's not always successful, particularly in older hens), by inserting the head into something that will hold it and then yanking hard in the opposite direction(I've used tailgate cables on the truck before), etc.
One lady here uses a BB or pellet gun as well and that seems pretty bloodless and quick also.
Here's one way and I've used it with much success....the head only comes off in very young birds and will often do so in the broomstick method as well. This is the best one I've seen on the method I use...the quick up and down jerk, using the bird's own body weight and momentum to dislocate the C1-C2 joint. You'll see less post mortem flapping and movement when using the dislocation method, especially when done effectively.