Generally, Hanon and regular fingering both are played by classical pianists
, because both are great tools, but they are for different targets. I doubt Hanon considered those exercises for non-classical playing, since it was published in 1873. You can't easily play sequence moving stepwise in your improvisation and use classical fingering, also you can't play fast ascending scale through a few octaves without using classical fingerings.
In jazz, Hanon fingerings (or lack of it
) is essential, one of my favourite jazz piano teachers, which is a genius player, said: "if you always play this note with the same finger, then you always sound the same. I can tell what you will play next". If you don't understand this statement too literally, he really has his point.
I play Hanon every day in all keys
, I play mostly first 20 exercises, and that was enough for me for last 3 or 4 years. I don't have enough time to play everything, so usually I play one of variations:
1) a couple of exercises through all keys
2) first exercise in C, C# and F, second in F#, G, G# and so on through all exercises
But rhythmically I also play one of variations:
1) equal 8ths, metronome 4x slower (usually 22-25)
2) swing eights, metronome once a two bars, or 2&4, or just 2, or just 4
3) equal eights, but with accents on 1&3 or 2&4 or every fifth note (last one is more demanding so tempo is slower, like 20 or even 18, but at 18 it may be better to play 36 instead).
Especially accents every fifth note is very demanding, and you shouldn't overdo forcing only those 5th accents, because it requires slower tempo and may make your fingers stronger, but affect speed and fluency of playing. So alternating exercises is best way.
What I plan is playing Hanon on pentatonics, which I heard is good. Also in melodic scales, I also wonder if it's possible to play it in diminished and whole-tone scale.
EDIT: Here is a link to first 20 execrises
with my additions (short, original Hanon comments which exercise is for what)