We used a combination of sharps and flats because that's what most people who study music would use. You never find a piece of music in the key of Gb, it will be in F#.
IOW, people who have formal training use the following convention:
C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B
In my experience, when listing or notating the notes (listing and notating are different activities) in a chromatic scale there are two main conventions:
1. If it's a chromatic scale
notated in standard musical notation in the
context of a major or minor key, then some notes will be listed with a sharp or natural sign and some with a flat or natural sign depending on whether or not the scale is written in ascending or descending order.
E.g.
In the key of C major the typical way to
notate an ascending chromatic scale starting on C would be
C C# D D# E F F# G G# A A# B C
while a descending chromatic scale in C major, starting on C, would be notated as
C B Bb A Ab G Gb F E Eb D Db C
The same thing for a chromatic scale asc/desc in F# major would be:
F# Fx G# Gx A# B B# C# Cx D# Dx E# F#
and
F# E# E D# D C# C B A# A G# G F#
I.e. If the line is going up, the accidentals used for the non-diatonic tones are sharps or natural signs.
If the line is going down, then any non-diatonic notes are written with natural signs or flats.
2. If the object is to merely
list the notes of a chromatic scale in ascending order, usually starting on C, then all sharps or all flats are used for the notes that are not in the C major scale, as in the example above in C major.
Typically, if this listing is in ascending order, as is usually the case with a chromatic tuner, then all accidentals will be sharps.
3. If one is
notating an actual musical line in a key signature that requires some accidentals for some chromatic notes, then the convention again is to use sharps/natural signs for chromatic notes that resolve upwards and flats/natural signs for chromatic notes that resolve downwards.
E.g In C major:
E F D# E
E Eb D
Etc.
On my Axe-FXII's tuner the convention seems to be all flats.
Has this changed with the III?