I think it will be difficult to answer this question in exactly the way you framed it because these voice types are not something that exist in nature — they are categories conventionally agreed upon to describe a complicated natural phenomenon (or, I'd argue, confluence of phenomena). The best analogy I can think of is asking what percent of people are sprinters vs marathon runners — most people aren't particularly either.
The voice types are most developed in the Fach system in opera, and even that falls apart quickly: many roles can be assigned to different Fachs, many people's voices are hard to classify in a Fach (especially when you add their acting predilections into the mix), and some people can be trained to sing in very different Fachs (e.g. countertenors often also sing, not as high-ish tenors, but as basses). This system is also very different from the way voice parts are assigned in choirs, which itself changes in typology over time. (Choirs have less highly trained voices and generally have to make the people available sing the music at hand.)
My answer, then, is not from statistics, but from my subjective experience as a singer, conductor, and voice teacher:
- At a professional level, women with excellent extremely high ranges seem much, much more common than those with excellent extremely low ranges. (However, the economics of demand may play a role in terms of how many professionals the market can support: to my perception, the low alto voice part is not used to anything near its potential by either composers or those responsible for programing.)
- Singing men are generally less thick on the ground at all levels than singing women, but professionals tend to be high-ish baritones most often, then truly high tenors (though again, economics may come into play), and truly low basses least often.
- Most untrained singers have a small-ish range (maybe a bit more than the interval of a tenth 10th) at a moderate pitch level, with about an octave or so between a typical woman's range and a typical man's.
- Many of the women who fill (especially amateur) choirs' alto sections may in fact be more soprano-ish than some of the same choir's sopranos, for a variety of reasons:
- In the least ambitious choirs, the soprano will often be singing a clear melody while the alto sings harmony, which means the "altos" benefit much more from a modicum of musical training, since they need to read, remember, and maintain an independent line. Shrewd directors with few trained singers may therefore put more of the educated ones on alto, regardless of what they sound like.
- Vocal training greatly increases range: a professional soprano would be able to sing far lower, in most cases, than an alto-ish woman with no training or experience. Thus, women with some training would be better at singing both high notes and low notes, and some of those will be wanted in the alto section. This is especially true in church choirs of modest aspirations, where the "sopranos" (aside, perhaps, from a small group to sing descants) are almost exclusively singing the melodies of hymns, which are designed to fall within anyone's range.
- Some women (for reasons inscrutable to me) simply insist that they are sopranos.
- Male singers are harder to come by, and most of the untrained prefer a small, baritone-ish range. The consequences of going too high are audibly disastrous, where as going too low simply becomes inaudible. Many composers and arrangers work with these facts in mind. That said, in the most benighted choirs I have seen, a robust-sounding bass (but not at all low by a trained singer's standard) seems most common, followed by at least a singer or two who get get a high bit out well, and low bases (really any pitch below the bass-clef staff) least common of all.