Actually the correct reply to the question in the subject of the thread
Rakes....where do you place it after using it?
is surely
Where the club asks me to place it
I actually have no say in the matter
I don't think I've ever played a course which told me where to place a rake. Not being funny, I've genuinely never seen a sign, been told in the pro shop etc.
If I play at an away course I will replace the rake exactly where I found it unless there are stands or holders that should have been used but simply haven't been.