The first sentence is most correct, but in most sentences neither would be.
Commas in english, are used to separate phrased idea's. Or another way of looking at it - whenever you would deliberately pause when speaking a sentence (not an incidental pause because you run out of breath), you would insert a comma.
'Her supervisor Susan' is really just the subject of the topic.
'Her supervisor Susan wanted to implement a new procedure'
'As her Supervisor, Susan could tell her (the employee) to serve customers"
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.