Well, I hate to say that I'm "for" euthanasia, because that simply sounds too morbid. But I'm definitely not against it, and I do think there is a place for it.
I don't think we have to delve into the reasons of medical issues, they are pretty self-explanatory I think.
Behaviourally, I think there are many situations where euthanization is the kindest, most humane method of easing an animal's pain. As Calliecritters mentioned, some dogs are just living, literally, a life of torment. Even with drugs, some dogs just cannot thrive in this world any longer. Some dogs are just beyond help, and in some cases where help might be a possibility, help would be so stressful itself, and the time it would take would be so vast, that the animal would still be suffering for a long time.
Then there are those animals who have very serious problems, but whose problems could be helped by the "right" people. The unfortunate problem is that there are not enough "right people" in this world that are able to help those animals. The number of people actually qualified to help these dogs (and by qualified I'm not even talking about credentials here, but simply people who CAN help in some way or another) is so low that it's just not realistically feasible to say "well somebody can solve this issue". Yes, somebody out there in the world might be qualified, but if that person is in across the country, in another country, or hundreds of miles away, unfortunately it's just not a plausible situation.
I'm a person who is very unsupportive of no-kill shelters (true no-kill shelters, not the ones that move dogs around so that they actually are euthanized at some point). I understand the compassion behind it, I do, but I can't in good faith support the keeping of animals for years, sometimes their entire lives, in a 6 x 6 prison cell, with perhaps one, maybe two toys, with a walk and attention twice per day, and very often no contact with other dogs (some shelters do, but the majority don't seem to do a lot of regular interactions between dogs). Dogs have gone basically clinically insane living in these quarters, and simply living in these quarters for years on end, has ruined any chance for that animal to lead a normal life, even if it went into the shelter being a normal dog.
If you look at the human equivalent to these shelters - prisons. Suicide rates are higher in prisons than the rest of the population as a whole. In fact, suicide is the leading cause of death in prisons. Now, of course this is a HUGELY complex issue that is not solely attributed to being in prison, but there is a portion of the problem attributed to life in a prison. And to think that prisoners often have a LOT more freedom than shelter dogs do within the environment.
As to what constitutes a dog being eligible for euthanasia, it's so individual I'm not sure we can really ever properly define it. What is a reason for me is not necessarily a reason for somebody else. The reason somebody else would do it, may be a reason I would never even consider doing it for. I may have been able to deal with the problem they are, but that doesn't mean they can, or that they should have to. And I don't think it's anybody's place to criticize somebody else for trying to do the right thing, even if you don't feel they are doing the right thing. It's just impossible to sometimes put yourself into another's shoes to know what they are feeling. I have the privelege (or unfortunate reality, depending on how you view it) to understand the range of emotions that people go through when dealing with problem dogs. Because I've been there. I AM there, and live with such dogs. And it's hard, it is really, really hard. Living with a problem dog really does change your entire life. So it's really not ever for me to judge when somebody decides they can't live that kind of life.
I think it would be interesting to hear of people sharing their bite stories, if they have bite stories. I've been bitten purposely three times in my life (two grooming separate dogs, another by a very anxious GSD that was at the boarding kennel I worked at - all unexpected that I couldn't have prepared for), and a couple of times accidentally breaking up dog fights. None serious, broken skin a few times but usually one or two holes with no requirement of stitches. The GSD incident didn't break the skin, but my arm from my wrist to my elbow was completely bruised with very deep bruising (not skin-level) and my muscles in my arm were very sore for two weeks because of it. But perhaps that's for another post.