Basically, I'd like to bring up the subject of women spreading the rumor that they were raped by some person, compared to the women who were actually raped and take the case to court.

I think a lot of MRA's are fighting against the false rumors and Feminists are taking it as us denying actual court cases or saying that the court cases are false accusations. It's very difficult to document rumors because they're just rumors. Statistics on these rumors are very hard to come by and I am unaware of any significant actions taken against the spread of rumors. These rumors can be devastating to a young person's reputation. No longer are they a random person on campus, they are "That guy people were saying raped some girl". The problem with the situation is that it's hard to prove that they started a rumor because when confronted by authorities, they just say they didn't start it and that it's untrue. Usually authorities isolate the person believed to have started the rumor from their friends so admitting that it wasn't true is never public. They can go back to whispering to their friends about it and based on our culture's "Fuck the Police" mentality, they just blame police corruption for the lack of discipline against said perpetrator. When their friends (who help perpetuate the rumor) are confronted by authorities, they seem to develop memory problems, can't exactly remember where they heard it and just state that everybody is saying it, that they must have over heard such things with people talking about it at lunch or some party.