I was curious so I looked it up - this is what I found as "humane" methods. It also lists the methods that are not recommended...
Site is about Culling Canaries:http://www.aaawebmaster.com/canary/culling.htm
Carbon dioxide is recommended by most animal rights activist groups and used by human societies around the world. CO2 is inexpensive but you have to make or buy a Euthanasia Chamber and have a way to fill the chamber with the gas. Other advantages: The rapid depressant, analgesic, and anesthetic effects of CO2 are well established. Carbon dioxide is readily available and can be purchased in compressed gas cylinders. Carbon dioxide is inexpensive, nonflammable, nonexplosive, and poses minimal hazard to personnel when used with properly designed equipment. This gas causes very little distress to the birds. A CO2 concentration of 60% to 70% with a 5-minute exposure time appears to be optimal. This method is recommended provided that the gas comes from a pressurized cylinder and not a chemical reaction or internal combustion engine exhaust.
Same as carbon dioxide only much more dangerous for the user. This gas is colorless and odorless and builds up in your system binding to hemoglobin and replacing oxygen in your blood. You can poison yourself as well as the birds if you don't have a proper setup. Use only with the proper setup and then with extreme caution.
Highly effective providing you dose correctly. Recommendations — In order of preference, halothane, enflurane, isoflurane, sevoflurane, methoxyflurane, and desflurane, with or without nitrous oxide. I have not found any of these compounds for sale without regulations. They may be difficult to obtain and can be rather expensive.
Nitrogen or Argon
These two inert gasses are relatively safe to use (non toxic in low concentrations). Using either of these two gasses can be stressful to the birds unless the chamber is filled quickly and has less than 2% oxygen. Birds should be sedated first. Although N2 and Ar are effective, other methods of euthanasia are preferable.
Dangerous but effective. Should have a consistent technique.
Quick but messy.
Intravenous injection is necessary for best results and requires trained personnel.
In conclusion, the best way to euthanize is to use Carbon Dioxide and a chamber.