Recently I purchased this game, and I enjoyed playing it, and i feel that charts for TSC may be an option. One reason that it wasn't added was because it was a mobile game. However since then, charts for both S1-13 and S2-13 have been added, so things on TSC have perhaps changed.
Charts for TSC would be as followed (and I'm posting this myself as i feel differently about what DarkAura suggested, and I shall give my reasons).
The game consists of a story mode, and an arcade mode, story mode has trackable stats. Aracde mode is an endless mode where the stage is random so different each time, and you have to try and get as high as you can without dying.
It has been released on both IOS and Android systems.
The game has multiple characters, which you can buy. According to this
http://sonic.wikia.com/wiki/Sonic_Jump_(2012) they are different in some ways, however I personally feel that the only character that should be tracked is Sonic, for simplicity, as you have to purchase other characters as extras.
Trackable stats:
Times
Standard time achieved without using any powerups, so that includes a blast off (gives you a massive boost upwards at the start of the stage), a magnet (attracts rings), a bounce back (allows you to continue if you die), a bomb (destroys all visible enemies), a shield, ring time (turns enemies to rings). Any one of those powerups would give you an advantage, so to keep things simeple and equal, banning them would be the best option.
Score
This is calculated at the end of the stage, I haven't managed to work out how it works yet though. Again powerups should be banned.
Rings
Shouldn't be tracked in my opinion, as stages contain an item box, which gives you a random item, one of those items being ring time, this adds incredible inconsistency as it is RNG, people may disagree with me on this matter, but the fact that you cannot consistently get ring time makes me feel that ring charts shouldn't be included.
EDIT: Bosses (Forgot this)
You have to defeat the boss, whilst climbing upwards through an endless level, his attack pattern happens regardless of where you are on the ascent, you can get unmitied rings, therefore an unlmited score, so just time should be tracked for this.
Stages:
with added suggested stats
Green Hill:
Act 1: Times, Score
Act 2: Times, Score
Act 3: Times, Score
Act 4: Times, Score
Act 5: Times, Score
Act 7: Times, Score
Act 8: Times, Score
Act 9: Times, Score
Act 10: Times, Score
Act 11: Times, Score
Boss (Act 6): Times
Boss (Act 12): Times
Mountain:
Act 1: Times, Score
Act 2: Times, Score
Act 3: Times, Score
Act 4: Times, Score
Act 5: Times, Score
Act 7: Times, Score
Act 8: Times, Score
Act 9: Times, Score
Act 10: Times, Score
Act 11: Times, Score
Boss (Act 6): Times
Boss (Act 12): Times
Jungle:
Act 1: Times, Score
Act 2: Times, Score
Act 3: Times, Score
Act 4: Times, Score
Act 5: Times, Score
Act 7: Times, Score
Act 8: Times, Score
Act 9: Times, Score
Act 10: Times, Score
Act 11: Times, Score
Boss (Act 6): Times
Boss (Act 12): Times
Blue Sky:
Act 1: Times, Score
Act 2: Times, Score
Act 3: Times, Score
Act 4: Times, Score
Act 5: Times, Score
Act 7: Times, Score
Act 8: Times, Score
Act 9: Times, Score
Act 10: Times, Score
Act 11: Times, Score
Boss (Act 6): Times
Boss (Act 12): Times
A stage looking like planet wisp is coming soon, and it's been that way for quite some time, if a new zone does get released, then as Parax said, new divisions could be added without too much trouble, however from what I can tell, Sonic Jump doesn't get much updating nowadays.
Hope I covered everything. :V