When should I use each of the collocations "knowledgeable about", "knowledgeable on", and "knowledgeable in"?
If you must use knowledgeable, then
are your best choices. Knowledgeable on is not correct. On implies that knowledge is a physical object that can have a physical position relative to something. For example, the following phrases use prepositions that commonly apply to physical objects:
These phrases don't make sense.
If you remove the -able suffix from knowledgeable, the phrases can be used in the following manner:
He has great knowledge in the field of Mathematics.
She possesses significant knowledge about Geology.
The NSA has voluminous knowledge of our calling habits.
Removing the -able suffix allows you to describe the knowledge involved in a more specific manner.
All are correct; when we think about knowledge, there are books on, about and in things, like a book on arithmetic, or a book about medical science, a book in music and performing arts, etc.
Knowledge is a notion broader than a book. You have a few examples here
When you are not sure about a collocation, you may look it up over the Internet, in double quotes (""), to get the exact phrase. For example,
Then, consider the context.
I hope this can help. :)