According to this Reddit post, the numbers are roughly as follows:
Men: 20% Bass / 45% Baritone / 35% Tenor
Women: 15% Contralto / 35% Mezzo / 50% Soprano
For males, the ordering implied by these numbers agree with this statement in Merriam-Webster, which is listed under the entry for "baritone":
baritone In vocal music, the voice or register between bass and tenor, the most common category of male voice.
For females, the ordering implied by these numbers agree with this statement from Wikipedia:
The lyric soprano is the most common female singing voice.2
2: Aronson, Arnold Elvin; Bless, Diane M. (2009). Clinical Voice Disorders (4th ed.). New York, NY: Thieme Medical Publishers. p. 278. ISBN 978-1-58890-662-5.
Another user in that same Reddit thread states:
In The Diagnosis and Correction of Vocal Faults, James McKinney asserts that voice types are roughly distributed on a sort of bell curve, with most people falling into the middle voice types (mezzo/baritone), which lines up pretty well with my experience.
The graph below is taken from a study on the frequency of someone's natural speaking voice and it seems to provide at least some sort of evidence that might lend support for the claim that a bell curve distribution exists for vocal classifications. I read about this study on this page, which is also where the graph is from:
Given the information in the more credible sources, I am probably inclined to trust the rough numbers from the Reddit post. Glancing through the user TotoCotogni's other posts, he seems like a relatively credible source, as the Internet goes for hard-to-answer questions.
Note: this same question was asked in another music.SE question, but it doesn't look like a numerical answer was ever given.