The murder in this movie(!) is...
I know that the movie was not supposed to show any murder considering the true story behind it, but there is one person that was not proven innocent at all in this movie: It's Hyeon-gyu Park, the last suspect. Let me explain why...
The false proof
The proof provided in the movie to clear of Hyeon-gyu is that the semen on the clothes of the 3rd victim does not match with his semen. But that actually just proves that he did not ejaculate on her. Considering the other victims, the murderer never left (verifiable) evidence to begin with.
The unlikely coincidence
I wonder why the movie never told the audience why Hyeon-gyu Park is requesting that specific song on rainy evenings and only on rainy evenings on specific days that somehow always matched the days of murder. It's very unlikely that the murderer did not know about the requests. When Hyeon-gyu became suspected and put under observation, the 4th and last victim in the movie was killed with neither the song nor the rain taking place.
The leading matches
When the inspectors visit Hyeon-gyu's apartment, you can see a mobile radio device and a photo album showing Hyeon-gyu on a tree (the murder is shown on a tree right before the 4th victim being attacked). Many photos of the album also show a near friend - this may be connected to Baek Kwang-ho, who mentioned being burned in his youth by the murderer.
The interrogation
Hyeon-gyu has soft hands as described by the victim in the detached house. When he returned from military service, the murder began (the time without women around may have released his desire to rape women and get rid of them afterwards -> motive). He lies to the inspectors for no reason (he knew the song was played and claimed he was home listening to the radio till the end, but he obviously didn't listen till the end). During the whole interrogation, Hyeon-gyu did not object naturally, but reacted to the provocational statements.
The train
Did you notice that a train connects Baek Kwang-ho and Hyeon-gyu Park? Both scenes threaten them, while the witness is killed and the murder lives on.
But then again: This does only show that Hyeon-gyu Park is likely to be the murderer in this movie. ;)