Depends on locale. In Germany something like 90% of murder cases are solved/cleared.
In the U.S., I suspect a majority of the murders technically unsolved by police are cases where the identity of the perpetrators is somewhat of an open secret within communities that don't trust law enforcement (and LE similarly has little interest in working with them either.)
>In Germany something like 90% of murder cases are solved
You must watch out when reading the German crime statistics. "Solved" which is marked as "aufgeklärt" in those statistics just means that a suspect has been named. Not that someone actually did it/has been sentenced for the crime.
Surely it's pretty common everywhere to have at some point a suspect ('solved!') who is then released, because you lack evidence, realise it's not them, whatever. A suspect isn't necessarily convicted even if you do ultimately convict someone.
Turns out it was someone else, and you convict that other person. You thought you had them, were wrong, but did then ultimately solve the case.
It happens loads too, frequently in high profile stuff on the news they'll have a suspect who's somehow close to it, arrest them, but then they're released once satisfied with their allibi or whatever.
Or most murder investigations are (by definition) incompetent.
Or (more likely): The old idiom quoted above is stupid and useless. (That it presumes that murdering and getting away with it is somehow a noble or esteemed deed should be damning enough.)
There’s no money or benefits in solving crimes. It could be done easily in many cases but nobody cares about certain people like gang members. Lots of cases where the murderer tells everyone but nobody cares.
The 4th option they may appear to propose suggests that murder investigators don't get paid -- neither in money, nor in benefits.
So, to that end: As far as I know, that's not usually the case with government employees, and it is always actionable when it does happen to be the case.