I had to get one many years ago, but my memory is fuzzy on this.
It was called a background check (could have been known as
something else but the Americans call it such so that's what I
remember i as). From the time of application to receipt, it took
about 2 months. I remember having to go to a police station in
Chinatown to get this done. I can't remember if I paid a fee or
not.
We were moving from Japan (where I also had to get one done, took
about 2 months too) to get stationed in Hawaii. I was told that I
couldn't move there directly without getting that Singapore Police
background check thingy. And that I had to be in Singapore to apply
for it (rules back then, may differ now). So I had to spend 3
months in Singapore waiting for the paperwork before I could move.
There were many other things I had to get done too. $$$$$$

I do know that if you've lived in one country for more than 3
years, you will need a background check from each of those
countries. (Those were the rules then, don't know about now.)
HTHs.