>99% of the current music on planet Earth is tuned to 12-EDO, but it wasn't always that way. Ages ago, different cultures were figuring out how to tune musical instruments in different ways, and came up with some pretty cool methods to make instruments sound in tune, whether playing on their own or with singers.
For people immersed in western culture, it might seem like a silly music fact now, that there are 12 different notes in an octave, but this was not always known, in fact, at some point, it was invented, and there were competing ideas about how many notes there ought to be that lost out. In some other cultures, there are, in fact, not exactly 12 notes from which to choose. For example, classical Indian music uses 22 notes based off of consonant tones which happen to be simple mathematical ratios from the lowest note, which is often used as a droning bass note. Within the idiom of music in the eastern world, there are 17 or 24 notes, which take the 12 western notes and add more in-between notes. In Polynesian and Indonesian cultural music, there are a smaller number of notes available.
For western culture, 12 notes was a good choice. With those 12 notes, the player of an instrument with a finite number of notes can get reasonably close to all of the important scales and modes. There aren't enough notes there to really get lost too frequently, and the scales sound in tune...ish.
So, fast forward to present day, and forget about the 12-EDO temperament that makes up for 99% of the music of current interest in the world, and what's left?
The biggest ones have to be the temperaments necessary to play the music of the near and middle east (maqam) and the music of India (raga).
Modern maqam in the Arabic-influenced world comprises of 24 equal divisions of the octave, or 24-EDO. If you are familiar with the modern temperament standard in Europe, China, and the Americas, 12-EDO, then you get 24-EDO by adding a note halfway between each note. Since the tuning is so widespread, with general use from Morocco to Afghanistan and beyond, the 24-EDO tuning set is the second most popular tuning worldwide.
The classical Indian tuning is the next most popular worldwide. Instead of using equal steps, the culture relies on shruti, which are unequal steps that are placed for the maximum usefulness in one key. The tonic note is called "sa," just like "do" in western culture (Solfege... "Do, a deer, a female deer, re a drop of golden sun..."). There is no key where a musician would use all 22 shruti, rather the musician makes up a scale out of all the shruti and then makes up a melody from the scale. This has an effect of very consistently pleasing music, but makes key changes very cumbersome, as the instrumentalist would have to retune all of the notes on the instrument to change key.
Then there are gamelan tunings, which are equally spaced notes, generally, but with a smaller selection of notes than 12, like 7 or even 5. The simpler scales lead to melodies that sound unusual to western ears, but are still melodic.
So, then what about experimental and unorthodox tunings. Well, that's my interest. Once I learned that there were other folks out there doing research on tuning instruments to have not-twelve notes, I immediately wondered which tuning systems those people tended to use. From my experience, 17-EDO is the most common tuning used by experimental musicians, followed by 19-EDO. After that, it really doesn't seem like there are a lot of people really congregating around any specific tuning systems. I've seen interest in 15-EDO, 22-EDO, and 31-EDO, but when I think of those tuning systems, it makes me think of one particular personality associated with that specific temperament, and then there is some discussion about that person's composition methods, and then some people who really like to try lots of different tuning systems will come up with some cool new music in those systems, but eventually move on to something else, and then subsequently something even more different after that. From the number of search engine hits and the amount of information on wikipedia and the xenharmonicon, 17-EDO is the most popular, followed by 19-EDO, and then everything else. It seems that 31-EDO has fewer search engine hits than other EDO's, but as you look at sources more focused on microtones, it becomes increasingly more popular, so, I think 31-EDO would be next after 19-EDO, and then 22-EDO after that, in terms of general popularity among microtonal enthusiasts.