I was told the following by an old chicken-farmer when I was young:
'humane' is our (humans) notion/idea of the least amount of pain/stress/fear (which we believe takes place in the brains).
He believed that the brain in decapitated heads (and broken necks) remain operational for about 30 seconds (probably based on well known guillotine stories where the victims blinked their eyes for 15 to 30 seconds, could answer questions (with eye-blinking) and make facial expressions (to the executioner)).
Quite recently, researchers found that the brains of decapitated rats remained conscious for about 3.7 seconds.
- He would smash the chickens with their heads against a wall (knocking them out) and then break their neck (or bleed them out, the drop in blood-pressure and lack of oxygen would keep them unconscious). He grabbed the bird by the legs (the knuckle-part of his index-finger was then near the anus of the chicken, which he therefore called his 'ass-finger', so the palm of his hand faced the chicken's head) and swung it hard in a circular motion (like a backhand in tennis). The centrifugal force ensured the back of the chicken's head was the first thing that hit the wall.
- (Small) injured birds (laying on the ground) got knocked out with a swift blow from a rock or log (which usually also smudged their entire head (between the ground and object) which was kind of an intended side-effect).
He learned this from his father, who in turn learned this from his father, etc.
While finding references to the 'lesson' (above) that I was taught (instead of just posting something I once learned), I found the following blog describing a similar methodology, they swung the bird overhead:
They also note that they learned this technique from another farmer and that they:
once saw an Australian Aboriginal women do this with monitor lizards in the Outback. She drug it from its hole (after tracking it) by the tail and swung it overhead–exactly as I’m doing in the above photo–and brought its head down over a rock.
Ironically (for this answer), I once had a vacation-job at a slaughter-house. The pigs and cows that entered there were knocked out by a 'non-penetrating captive bolt pistol'. Think of it like a one-handed single-shot mini 'jackhammer' (which is what we called the thing). Variations of this thing are also used for in-the-field euthanasia (of animals).
So, it appears (to me at least) that what the farmer taught and practiced might indeed be the best way to go about it (since the underlying idea is identical in all references and even cultures): knock 'm out (optional side-effect: destroy head/brain), then finish-off the bird properly, for example by breaking it's neck as described in other answers (just make sure it doesn't wake up with a splittin' headache added to it's list of ailments once you are gone..).
Follow the warning from the other answer: Don't do a half-assed job!!!
That being said, It appears there is no need to put the bird in a plastic bag and swing it, in fact: it would be hard to take proper aim making it a piñata instead of hitting the bird head-first.
EDIT: partially due to the picture, this answer might appear to mostly advertise 'swinging the bird', while it is more intended to explain why the seemingly barbaric method of 'bashing in the head' is often the best humane way for birds that can no longer run/fly away from you (and simple: just find a rock or log). Also, it might be safer for yourself (you don't know for example what kind of disease the bird carries).