"Knowledgeable about" vs. "knowledgeable on" vs. "knowledgeable in"

If you must use knowledgeable, then

knowledgeable in

knowledgeable about

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:

knowledgeable under

knowledgeable over

knowledgeable beside

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.