I belong to the school of thought that treats all animals in the compound as pets. Sometimes pets including cats and dogs suffer from debilitating conditions and it seems that the only recourse is euthanasia. Their continued existence can sometimes become a form of torture. I am referring to such conditions as melanomas and certain congenital abnormalities. One of these situations is when a pet becomes a threat like when a dog attacks and kills or mains a child due to reasons not well understood, that is the pet becomes a clear and present danger. There is also the situation where a pet is killed for consumption to feed other carnivorous pets or for the palate, cases of extreme morbidity due to old age, the list can go on. One point to note is that such killings should be done only if its absolutely necessary. What are recommended humane techniques of dispensing with such pets if it has to be done?
In almost every case my opinion is the humane technique is to take the animal to a person trained in humane techniques. That is: a veterinarian. I just took my rat (with advanced mammary tumor) to be euthanized. $60 including cremation. That's a small price to pay for her to go peacefully with an overdose or anesthetic rather than whatever I might try.
If people are experienced in humane techniques for whatever reason, and have access to the appropriate tool or drugs, then fine. But otherwise, leave it to the experts. The only animals I euthanise myself are fish and I use legally purchased anesthetic finquel (overdose). As a general rule I don't think anyone should use a method not approved by the AVMA Guidelines.
Also various laws apply. In most US states you can kill you own animal if you do it humanely, but killing any other person's animal may be illegal.
As I had posted in another thread, my father was a trainer of race horses and he had so many horses in his stable. One cause of death of a race horse is broken leg. A race horse with a broken leg is shot on the head with a gun. My father had done that so many times - when his horse would figure in an accident. He said that a broken leg is so painful so he would end the agony of the horse. But I haven't seen that killing for I was still a girl and I'm sure my father wouldn't approve of it. But when he comes home at night, he would mention to us that one of his horses died. And when the cause was a broken leg, we already know that he had used his gun.