My personal opinions for when it might be appropriate to consider euthanizing a dog for aggression:
-- If the dog has seriously injured a person ("serious injury" to me means that the victim had to go to the hospital and/or get a significant number of stitches for the injury) under inappropriate circumstances (by which I mean that if one of my dogs bites a mugger who is trying to rob me, that is absolutely fine and dandy as far as I'm concerned).
-- If the dog has not yet seriously injured someone but the dog's aggression is getting progressively worse (for example, moving from "teeth on skin, no pressure" to puncture wounds) AND the owner cannot or will not manage the dog.
Some of these are gray areas where the dog might be okay in other hands, but the owner cannot or will not relinquish the dog (and even if the owner wants to hand the dog over to someone else, there is a very finite list of people and places willing to take on such dogs, and none of the good ones are cheap), and ultimately IMO the safety of the public overrides that dog's interests.
-- If the dog's aggression is rooted in other medical issues and those medical issues cannot be resolved. I include both physical illness and mental illness in this category. If a dog is in so much misery that his pain is coming out as aggression, it may be kindest just to let that dog go.
ARCHX TDCH Pongu the Insane, CD-C, RL1X3, RL2X2, RL3X (GSD mix, b. Apr 2010)
Crookytail the Tigerwuff, RL1, NTD (Akita mix, b. Jan 2011)