What is the difference in meaning of "frustrated at" and "frustrated with" used in sentences like
He is frustrated with me
He is frustrated at me
When is each of these sentences above are used?
Strictly speaking there isn't much of a difference between the two. If you interchanged one for the other no one would correct you. That being said, when placed side-by-side, there is a subtle distinction between them.
Being frustrated "at" is directing the emotions toward the recipient. Being frustrated "with" is directing the emotions elsewhere. It is a deflection so small that it requires a heavy emphasis to even notice it.
The primary use for the distinction is to associate the perceived solution being something inclusive or exclusive. If someone is frustrated at you they probably think the solution is to remove you from the equation. If someone is frustrated with you they are more likely to think of the situation as the problem — not you personally.
It is little more obvious with other emotions:
The laughter at/with example is the most common.
These types of prepositional emphasis are relatively common when trying to describe particularly emotional acts:
First of all, it's the difference in meaning between X and Y, not of X and Y. Prepositions are governed by what they modify, and difference governs between.
Second, you have both an is and an are in this question, so the grammar needs fixing. A much simpler question would be How are these used?
Also, you're asking for something that doesn't really exist -- lists of all situations in which one preposition is used, but the other isn't. Which assumes that there are such lists, that we know them, and (most important) that there is a difference between the two forms in every case. This is all gratuitous assumption.
Returning to the two prepositions are concerned with the predicate frustrated, normally it takes with. However, most predicates referring to negative emotional display take at to indicate the recipient of the display.
These seem to be directional uses, where the focus is on the display and its recipient, whereas the constructions with with seem to refer more to the emotions and their cause, which (or more likely who) is represented with with.
because mad does not take with. As I said, preposition use, like almost everything else in a sentence, is governed by the predicate, which is the center and most important part of any sentence. As I like to put it, "Verbs Have More Fun."