Prosecutions for falsely reporting rape are at least as common as perjury convictions (and usually don't count as perjury since the initial report is rarely made under oath), and even when charges are brought by prosecutors involving false statements made under oath, prosecutors tend to favor lesser misdemeanor false reporting charges over perjury charges. See, e.g., cases with news reports of such cases in Colorado and Wyoming on May 24, 2021, September 11, 2015, August 29, 2014, and March 18, 2008.
This isn't to say that these cases are terribly common although they tend to generate headlines when they are brought. The State of Colorado commenced 1,801 felony sex offense cases in the 2021 fiscal year, for example, which was not atypical, and false reporting of sex offense cases are brought in Colorado maybe once every year or two.
The conviction rate in sex offense cases that are prosecuted isn't 100%, but something on the order of 90%-95% of sex offense cases result in a guilty plea, and well over half of the remaining cases result in convictions at trial. As an order of magnitude estimate, perhaps one in fifty to one in two hundred cases where sex offense charges are pursued, but there is not a conviction, gives rise to false reporting charges against the alleged victim.
Many acquittals and dismissals of charges that do occur are best characterized as cases where there is a reasonable doubt because jurors believe that it is reasonably possible that there may have been a good faith witness misidentification, or because charges were dismissed because a confession or evidence obtained in a search was unlawfully obtained and had to be suppressed. It would be very rare for a defendant to be acquitted (after a judge in a preliminary hearing found that probable cause was present) because the jury believed that the testimony of a victim was believed to be intentionally false, and there is no way to tell from the verdict itself that the jury reached this conclusion.
The problematic aspect of charges of false reporting of sex offenses is the there have been famous instances of women being convicted for falsely reporting rape (see also here focusing on a different case), only to subsequently have the allegations for which the victims were punished confirmed to be true with DNA and other evidence.
The number of cases where true allegations of sex offenses are made but not pursued because law enforcement finds the allegations to not be credible, almost surely greatly exceeds the number of cases where false reports of sex offenses are made to police, although this ratio varies greatly from one police department to another based upon the institutional culture of the police department in question.