Her supervisor Susan, wanted to ...
Her supervisor, Susan wanted to ...
which one is correct?
These are both correct:
Her supervisor Susan wanted to ...
Her supervisor, Susan, wanted to ...
But you must either use two commas or none here.
Sometimes the appositive and the word it identifies are so closely related that the comma can be omitted, as in "His wife Eleanor suddenly decided to open her own business." We could argue that the name "Eleanor" is not essential to the meaning of the sentence (assuming he has only one wife), and that would suggest that we can put commas both before and after the name (and that would, indeed, be correct). But "his wife" and "Eleanor" are so close that we can regard the entire phrase as one unit and leave out the commas.
This simply says that this is correct:
"His wife, Eleanor, suddenly decided to open her own business."
Leaving out the commas, because it causes no problem in understanding, is also fully acceptable:
"His wife Eleanor suddenly decided to open her own business."
You have a choice. If in doubt, use the commas. They are not wrong.