I'm afraid I've had limited success with the air rifle, but I think perhaps mine needs a new spring.
The broomstick method for a chicken is to hold the bird's feet, place its head on the floor, then put the broomstick on the base of the spine. Then you thank the bird (an inuit tradition which we now keep to!) apply downward pressure with both of your feet on the broomstick (one either side of the bird's neck), and pull upwards with your hands.
It's more than just pulling though. I've found that the round shape of the broomstick is important, since that seems to settle in between the bones, and allow for quick dislocation. It's not pleasant, but it's about the least unpleasant way I've yet found.
I can also do it this way for turkeys, but find geese much much harder. I believe the recommended method for them is to hold them round the body, put the neck under the stick, stand on it, and then stand up. It's easier said than done though, and I'll be honest, I've been putting off killing our remaining geese because I'm not yet 100% sure of the best way to do it. Any better ideas folks?