Why are there no commas before some prepositional phrases and not others?

Question

I was taught that there shouldn't be any commas before or after prepositions. Why is that in some sentences such as sentence 1 below, a comma before "of" is permitted?

Detective Polly found that, of the hundreds of cases of malaria near the river, all but three involved near the contaminated pump.

Or for sentence 2, why do we have to have a comma between "program" and "from"?

They now operate from 12 to 2pm, and the government funds every aspect of the program, from the jerseys to the coaches.

Or for this sentence...why is that we can't have a comma between "VP" and "about"?

I signaled to the VP about the wrinkle on his tie.

Answer 1

Detective Polly found that, of the hundreds of cases of malaria near the river, all but three involved near the contaminated pump.

... because "of the hundreds of cases of malaria near the river" has been moved out its natural position as part of the subject of "involved", to a more visible position; here is the natural version:

Detective Polly found that all but three of the hundreds of cases of malaria near the river involved the contaminated pump.

And with the next sentence:

They now operate from 12 to 2pm, and the government funds every aspect of the program, from the jerseys to the coaches.

... because "from the jerseys to the coaches" is merely an afterthought, an adjunct, extra information extending the idea of "every aspect". An em-dash could be used there instead of a comma.

And with the next sentence:

signaled to the VP about the wrinkle on his tie.

... because "about the wrinkle on his tie" is integral to the purpose of the signaling, not merely extra information.

A prepositional phrase introduced by about completes the sense of many verbs, for example:

To signal someone about something...

To tell someone about something...

To question someone about something...