According to Cait McKeown (a National Mice Club (UK) member and judge), the most humane method of mouse euthanasia is chloroform. Unfortunately, the chemical is hazardous (even deadly in high dosage) to humans as well as mice, so it's difficult to obtain. Also impractical for most people, veterinarians sometimes use halothane or another anesthetic gas. With these methods, the mouse becomes unconscious without pain before death.
Other methods, such as breaking the neck (cervical dislocation), decapitation, drowning, and freezing are painful for the mouse. This may or may not be an issue for you since these are not pet mice, but most people will have a difficult time in execution. The "quick" methods might not be so quick if you make a mistake.
Perhaps the best method is a CO2 chamber such as used by herpetologists before freezing rodents for food. The cheapest source of concentrated carbon dioxide is dry ice, but the National Institutes of Health (NIH) recommends compressed CO2 gas in cylinders [PDF] in order to control the inflow of gas. Their instructions are:
Without pre-charging the chamber, place the animal(s) in the chamber and introduce 100% carbon dioxide. A fill rate of about 10% to 30% of the chamber volume per minute with carbon dioxide, added to the existing air in the chamber should be appropriate to achieve a balanced gas mixture to fulfill the objective of rapid unconsciousness with minimal distress to the animals. (Example for a 10-liter volume chamber, use a flow rate of 1 to 3 liter(s) per minute.) Sudden exposure of conscious animals to carbon dioxide concentrations of 70% or greater has been shown to be distressful.
If you regularly use sticky traps to control mice, you might consider building CO2 chamber (there are plenty of designs out there). But I personally prefer using snap traps which provide a much quicker death and are easy to dispose of.