I grew up on a farm and my Dad 'educated' me in the "Rabbit Punch" and he called it. I've used it several times on mixy'd rabbits without too much trouble:

1. Pick the rabbit up by it's back legs
2. The rabbits ears will drop down with gravity exposing the back of the neck
3. Use a solid chopping motion following the angle of the ear to the back of the neck to instantly despatch the rabbit...
4. If you are unsure you can easily give a second chop.
5. Be aware that nerves sometimes kick in which can be unpleasant but are unavoidable - the rabbit is dead but its musculature is shutting down.

If you come across a bird with a broken wing the best way is to find a fence or an object with a sharp edge.

1. Hold the bird over the wings and around the body breast up. The bird will generally hold it's own head up.
2. Strike the back of the neck solidly on the edge to again break the neck.
3. The head should now hang limply, but again there may be some nerve action.

The holding by the head and wrist flicking option works with birds and rabbits but be aware with birds that the whole head may come off! Just don't be shocked if it does!!!
Hope this helps!

Thanks Morganmarbles - all useful information

At one time I would have been appalled at this thread!

Now, however, I have a whippet pup and although I would prefer her never to catch anything, I am realistic to know that she has a high prey drive and it is more than likely that it will happen unless I keep her muzzled when she off the lead.

Thank you for the info here - I will bear it in mind.