some gin in a birds drinking water will send it to sleep from which it will not return..... apparently.

alternatively, taking a pair of dog nail clippers (the pincer type not the guillotine type), and with one swift snip round the neck at the base of the skull. this may sound horrific and i can understand most could never bring themselves to do that. it is swift, and only a split second of pain.

as for taking it to a vet being the most humane thing, the stress of being caught up and taken to a vet and likely havng to sit in a waiting room for god knows how long with the sounds of other animals all around and then being handled by a stranger (in my experiance, alot of vets havent a clue hoe to handle birds properly and cause them more stress), then jabbed in the leg with a needle, a needle which we may think is fine but to a tiny bird must be like a javalin. and then taking a minute or so to die as the drugs take effect..... and be charged a fortune to boot!
humane?.... hmmmmm....