Note: This only applies for cases where we don't know they're dealing with a psychopath from the start.
Times where the killer was seen in the wrong from the get go/they might have broken down but there's no way their motive can be seen as sympathetic:
One more note though, there's a fairly recent case in the manga that I won't spoil, but there's a murder at a hospital, and when you find out what the motive is, it IMO is one of the biggest WTF in terms of DC culprit motives in the sense of I can't tell whether this means we should feel bad for the killer

