Apologies for what's going to be a nasty-sounding post. But if you raise chickens for meat and eggs as well as pets/hobby, you have to know how to dispatch.
A lot of people lately seem inclined to take their chickens to a vet. I've done this in the past myself, though only once — the cost was just plain prohibitive. I wish it wasn't. There are times when I've been unable to dispatch a sick bird, but unable to afford a vet, and the resulting death has been appalling. Later I've castigated myself for being weak. Once I even made my partner (who is not into chickens!) put a bird down for me, the ultimate in weakness. But dispatching something you've cared for and nurtured just doesn't feel natural.
And yet somehow I've gotten over the worst of all that. Yes, we're taught that animal cruelty is fiendish, but we're also deluded into thinking supermarket food hasn't suffered. Or we're taught to shrug and accept that some degree of suffering is okay if it's feeding the masses. Given the mainstream chicken industry's reliance on artificial feedstuffs and medications (a necessary reliance in that profit-driven mass-scale system), it's time to face the facts of where meat comes from. And the simplest fact of all is that it involves death.
With that preamble over, here is my general 'how to'.
First, some points. You don't want to use toxic chemicals to 'put a bird to sleep'. You want to bleed the carcass so the meat stays fresh for long enough to relax from rigor mortis and be good to eat. (Typically rigor mortis passes in 24-48 hours while in the fridge.) You want to kill a bird with as little stress as possible so the creature doesn't suffer and also so the meat isn't tough.
A lot of people use the axe, and in general it's very sure. I suspect the weight of it helps stop the person wielding it from baulking at the last instant (a thing that's very natural for anyone brought up to be kind). However it's messy, and in practice that method of killing sets the feathers hard.
American home growers seem to prefer debraining and throat cutting, and debraining (or sticking) seems to be quick as well as having the benefit of loosening feathers. However it's hard to go from debraining to throat cutting swiftly enough to get most of the blood pumped out while the heart is still beating, and it's messy. (You also need a killing cone.) Europeans tend to prefer the neck dislocation method, which if done properly results in the bleed-out being into the neck cavity and also destroys the brain at the moment of dislocation, but doesn't involve a two step process (so I feel it's quicker).
As you can probably guess, I don't particularly like the debraining method, for various reasons (mouths are sensitive... I can't bring myself to stick a knife in there to reach the cleft that leads into the brain); but as it happens I also don't have the arm strength to do the classic European method of dispatching a large cockerel by hand. For this reason I use a modified neck dislocation method that doesn't require great arm strength.
This is done with the bird calmly and gently placed on the ground on its chest. A metal rod (or broomstick, though the thinner and harder the rod the better) is placed, again gently, over the bird's neck just behind the skull, laying cross-ways. There should be no pain or stress at this point, and all the chickens I've done it to seem relaxed, especially if they're used to being handled. Holding the legs, the handler steps down firmly (but not trying to crush the neck!) on the rod and in the same instant pulls upward hard on the legs. The neck quickly dislocates just behind the head and the bird bleeds out into the cavity.
I've done it this way many times, and while I can't say it's painless, I do know that it's very quick and sure.
Now that I've spelled it out, I need to take a break... There's nothing enjoyable about killing. But there's nothing about supermarket meat that shows respect for life, either. Sometimes you just have to grit your teeth and do it.