The rates of false rape accusations are the same as any other crime - between 5-8% depending on the study.
That 5-8% is the percentage of reported rapes that are proven to be false or for which not enough evidence could be gathered to build a court-worthy case. Depending on the laws where you live, "legitimate rape" (ugh I hate that phrase) could be dismissed if the perpetrator didn't use violence, if the victim was too intoxicated to give an accurate account, if too much time had passed to collect viable evidence, etc. Just because a rape accusation is made and then dismissed doesn't necessarily mean the accusation was false, just that no one could prove it in a court of law.
Rape only has about a 40% reportage rate and over 95% of those accused will never spend any time in jail. There has to be some truly shitty planetary alignment to get someone falsely accused of rape imprisoned.
One of the reasons so many rapes go unreported is because of shit like these threads that automatically assume the accuser is some lying bitch out for vengeance or covering up "regrettable sex" or whatever. No one wants to go to the police or even tell their friends if they're afraid no one will believe them. When you make a big deal about false accusations, or when you try to pass harsh punishments for those who make accusations that can't be proven (see second bullet), you make victims afraid to come forward. False rape allegations hurt future victims, too. Every time one makes a headline, it makes it that much harder for future victims to be believed because the false allegations stand out in the popular consciousness.
Rape tends to happen between two people behind closed doors. Don't presume to understand "what really happened" when your buddy says he was falsely accused or whatever. Look at the percentages and ask yourself if your buddy is really such a special fucking snowflake or if there's actually a wee little chance the accuser is telling the truth.
Studies of men on college campuses have revealed that roughly 1 in 25 men will admit to behaviors that would be legally classified as rape - but they won't admit it if the word "rape" is actually used. They'll say, "Yeah, I used violence or the threat of violence to get a woman to have sex with me," and out of the other side of their mouths will say, "But I'm not a rapist." If someone you know denies being a rapist, that might just mean they don't consider what they did to be rape when legally and morally it is.
About that 1 in 25 - in the US, that's 6 million men. That's more than the number of Jewish people, or single mothers, or postal workers, or grocery clerks. How many of those demographics do you interact with on a daily basis? Because you've probably met more rapists. Think of how many men you know, and then realize that 1 in 25 of them is a rapist. There were probably 6-8 rapists in my high school class. How many were in yours? How many are in your frat or your classroom right now?
Think of how many women you know, and realize that 1 in 6 has been or will be raped in her lifetime (a conservative estimate; some researchers place that number as high as 1 in 3). 1 in 6 women will be raped, but only 1 in 25 men will rape. Those rapists in your classroom have probably done it more than once, and will continue to as long as you dismiss their behavior as "that bitch made it up."
3% of American men (1 in 33) have experienced attempted or completed rape in their lifetimes (comprising about 10% of all rape victims), and 21% will experience sexual violence other than rape. In cases of male rape, 6.5% of perpetrators are female. So as a man, you're more likely to be raped or sexually abused by another man than you are to be falsely accused of rape.
Sources: https://www.rainn.org/get-information/statistics/sexual-assault-victims
http://thehathorlegacy.com/rape-statistics/
http://thescarletwoman.tumblr.com/post/22768765527/one-in-four-or-5-men-are-not-rapists-thats-the
http://www.newsweek.com/campus-rapists-and-semantics-297463 (this more about men who will/want to rape but won't call it that)
http://america.aljazeera.com/watch/shows/america-tonight/america-tonight-blog/2013/10/28/serial-rapists-commit9of10campussexualassaultsresearchfinds.html (serial rapists on campus, including quotes from Lisak study)
http://amptoons.com/blog/2004/05/05/how-many-men-are-rapists/ (sites Koss study, which is where I got the 1 in 25)
http://www.davidlisak.com/wp-content/uploads/pdf/RepeatRapeinUndetectedRapists.pdf (Lisak study)
Koss study found 4.5% while Lisak found 6.4%. Both of these were studies of men on college campuses, so the general population might actually be higher or lower, but this is the best estimate anyone's got so far.
A UN study indicating that worldwide, the number of rapists in the population averages 1 in 4: http://www.thelancet.com/journals/langlo/article/PIIS2214-109X(13)70069-X/fulltext
2010 study indicating the instance of false rape (ie a person knowingly reporting a rape that did not occur with the express intention of causing harm to the accused) indicating 5.8% of rape reports are demonstrably false: http://www.icdv.idaho.gov/conference/handouts/False-Allegations.pdf
Another false rape study from the UK, 6% of allegations are false: http://www.cps.gov.uk/publications/research/perverting_course_of_justice_march_2013.pdf
Another from the US Bureau of Justice Statistics, 8% unfounded (includes false and those that cannot be proven): http://www.bjs.gov/content/pub/pdf/SOO.PDF
National Intimate Partner and Sexual Violence Study indicates 1 in 5 women have experienced rape or attempted rape: http://www.cdc.gov/violenceprevention/nisvs/infographic.html
2000 US DoJ report that fewer than 5% of attempted or completed rapes on college campuses are reported: https://www.ncjrs.gov/pdffiles1/nij/182369.pdf
Also, a CDC report on male rape determining that 21% heterosexual males have experienced sexual violence other than rape (more than twice the number that will ever be accused of rape!): http://www.cdc.gov/ViolencePrevention/pdf/NISVS_SOfindings.pdf