Why the real killer is .... *** major spoilers, or course ***
I think there are plenty of pointers to Catherine being the killer.
In my mind, the most important one is: Nick is able to get into Beth's apartment. He tells her something on the order of she should get her lock fixed.
The "evidence" that the police find in Beth's apartment following her death was planted there by Catherine. For one thing, it is very reminiscent of the evidence Catherine has been collecting about Nick.
Here are more reasons I think point to Catherine being the killer.
When Beth dies, she tells Nick she loves him. If Beth had actually been the obsessed one in college, she would probably have come clean at that moment and told Nick she hated him, loved Catherine, ha, ha, your partner is dead, etc.
This is of a very technical nature which might not even be germane because most screenplays might not look at this, but this one has enough attention to detail I think it could be relevant: we are told of no blood spatter on Beth. Even if she had worn the raincoat, there was enough blood when Gus was killed to have gotten into the opening over the face.
I haven't had the time to do this, but I bet if you analyze the movie you will find that every time someone is doing something that is a mystery, Catherine has time to do it. After Gus's murder, for instance, she has time to plant the evidence in Beth's apartment, or maybe already has. Also, didn't she at some point let herself into Nick's apartment? This could show she is capable of getting in anywhere.
Of course, there's the ice pick by the bed in the final scene, some pretty face-value pointers such as the book she wrote that mirrored her parents' death, etc. But one thing you might get a kick out of: on my second watching, I checked out the bod of the woman in the first scene and compared it to the bods on Catherine and Beth. You guessed it ... looked like Catherine's. So did what little we saw of the face, too. Try it yourself if you don't believe me. I usually don't do that sort of thing. But inquiring minds want to know.
Anyone else have anything to add, please do!