So I always wanted to compose a song in 31-TET inspired by a song I heard on musescore some months ago, the thing is that I didnt know how to notate in 31-TET, so obviously i want to learn.
A diatonic semitone ("between tones") is between two note names in the standard major scale, represented by B-C and E-F, and also by implication between F♯-G, A-B♭, etc.
A chromatic semitone ("colour", i.e. black/white on a keyboard), is what single accidentals do on the same note letter. C-C♯, E♭-E, etc.
The fundamental rule is: diatonic semitone + chromatic semitone = whole tone - simply think about C-C♯-D (or C-D♭-D as well).
Along with the fact that 31-EDO has a whole tone = 5/31 steps, diatonic semitone = 3/31 steps (5x5+2x3=31), this is enough to derive all notation.
A single accidental mark is therefore 5 - 3 = 2 steps, so the 5 steps between C and D can be written as C-D𝄫-C♯-D♭-C𝄪-D. Between a diatonic semitone, we have B-C♭-B♯-C.
Normal major/minor scales work the same way as in any meantone system, as long as you keep the fully spelled out accidentals when double or triple sharps/flats exist. If you play around, you can find the enharmonic equivalences of 31-EDO, for example B♯ = C = D𝄫♭ (compare B♯ = C = D𝄫).
*: "equal divisions of the octave" instead of just "equal temperament". See Bohlen-Pierce scale for why.
There are two alternative notations for 31-TET. One of them uses double sharps and double flats:
The other one uses half sharps and half flats. There's more detail at the Wikipedia page for 31-TET
Generally, you don't notate for 31-edo. Recall that we don't really notate for 12-edo either, otherwise there would be no such thing as enharmonics.
Rather, as long as you intend to use it for tonal 5-limit music, you should mostly notate for meantone tuning, which can apply to both 12-edo and 31-edo. I.e., you notate in diatonic tonality, just as you're used to.
That shifts the question to: how do these notes actually map to 31-edo? But because it's meantone, that's actually straightforward: you can construct everything from the knowledge that a perfect fifth has 18 steps and a major third 10. For example, to figure out how many steps there are between C and D♯, you note that you can stack a perfect fifth and two major thirds to get from one to the other, i.e. 18+10+10 = 38 steps, which mod 31 is 7 steps.
Of course, you do not get the same enharmonic equivalences as you would in 12-edo. (Note that the minor third has 8 steps in 31-edo, unlike our augmented second C-D♯!)
Another matter is when you want to make use of the 7-limit or odd-11-limit capabilities that 31-edo offers. This could be expressed with quarter-flats and -sharps, or you can again base it on just intonation, using a notation like Ben Johnston's.
You just notate with the usual manner of notation. The whole point of TET-31 is that your normal scale is represented by 5 whole steps of 5/31th of an octave and 2 half steps of 3/31th of an octave. An accidental causes a change of 2/31th of an octave, a double accidental twice that (so an accidental is "less" than a "natural" half step, and a natural whole step is larger than the difference by a double accidental, but smaller than the difference by two natural half steps).
The main difference is that there are no enharmonics: accidentals don't go as far as the natural distance.
Notation/transposition work in a rather mathematically pleasing coherent and regular manner: any scale interval of a particular kind (major/minor xth) is identical in width to any other of the same kind. Standard notation is fully adequate for the representation, and yet everything is built from 31 rather than 12 notes in an octave, all of them being accessible to notation.
There are various different conventions. One of the more common ones is to incorporate symbols for
(Images are screenshots from MuseScore)
Of course, to compose in any "style" (using the term loosely here), it's highly instructive to study the scores of existing pieces. That's particularly true in learning how to notate 31-tone music.
The Huygens-Fokker Foundation: Centre for Microtonal Music has a list of 31-tone compositions. The scores are likely all under copyright (short of a composer deciding to make one public domain), but some might be available at a university library. Also, articles about those composers or their compositions may include excerpts that would provide notation examples. For instance, the article "Six American Composers on Nonstandard Tunings" by Douglas Keislar (Perspectives of New Music Vol. 29, No. 1 [Winter, 1991], pp. 176-211) includes an excerpt from Joel Mandelbaum's Four Miniatures for Archiphone, which demonstrates with 31-tone scale notation.