I'm wondering if there's certain chromatic passing tones that are used more than others. Especially in classical or pop improvisation/composition. For example in the key of C, each of the five black keys on a piano could potentially be chromatic passing tones. But are there ones that are used more than others, and do you usually land on a chord tone from a higher or lower chromatic note?
The two most common chromatics in key C will be F♯ (to modulate to key G), and B♭ (to modulate or even get to F). However, neither is used here as a passing tone.
Some ideas: C♯ would lead either to D from C - almost like a leading note getting to Dm, or, it could be part of a tts going back to C - in which case it's more likely to be D♭.
E♭ will usually slide to E, especially in C major - a blues feel. Not so likely called D♯.
F♯/G♭ is the tritone of C, and is used both in jazz (F♯) and blues (G♭).
G♯ comes along as the leading note to A, whereas A♭ can be seen as the m3 of F, sliding in blues form to the M3, A.
A♯ can work as m3 of G, again as in blues, but if it was B♭, it could be the passing note from a descending chromatic line - C>B>B♭>A.
So, they all get used, given their own circumstances. I'm not going to count which is their order in a list, it's pointless, but given enough experience playing, it's apparent they all get used - a lot. It's sort of - in a key, the diatonic notes get used, but there's no reason not to bring the chromatics along to the party - after all, they're the colourful ones.
Generally speaking, in ascending passages, one uses a raised chromatic passing tone. In descending passages, one uses a lowered tone. In either case, the voice leading would be the most important criteria.
For example, (in C major) a scale passage G-A-B-C would use G# or A# (or both) if passing tones were inserted. A descending scale passage like F-E-D-C would use Eb or Db (or both).
As to which scale step is most often altered, that would be hard to say. Based on what I remember from playing and listening (note that the plural of anecdote is not data) is the raised 4th and raised second steps are most common in ascending passages; the raised fourth goes up to the dominant. The lowered seventh and lowered third I'd would guess for lowered passing tone. The lowered seventh helps make the tonic seventh or V/IV chord; the lowered third makes a temporary minor chord; both are common.