Which one is correct in American English? What is the best way to say that?
1) Please scratch the upper of my knee. 2) Please scratch above my knee.
(2) is reasonably correct. It could be more explicit by saying:
Please scratch the area above my knee.
In (1), both "upper" (adjective) and "of my knee" (prepositional phrase) have nothing to modify. The following would be grammatically correct:
Please scratch the upper part of my knee.
Which one you should use depends on where you actually want to be scratched.