I've had just over a week now to understand what can and can't be tracked properly in this game and I believe we can start setting up charts. First a little run down of each Category:
Times: In story or "normal play" mode there's only a countdown timer, and when you finish the stage it disappears and adds onto the score so this isn't a feasible way of tracking.
There's a "Time Attack" mode which should be used to obtain all Time records in this game as it tracks your time and also puts it on the global/national leaderboards if you're connected to the internet.
Rings: There are alot of them but there shouldn't really be any restrictions right now. I haven't come across any rings that respawn yet either. There is no tracker on the stage menu, though.
Scores: nowhere on the HUD is score tracked. The only place you see your score appear is at the end of the stage, but that's an accumulative score across the entire game so these can't be tracked.
Animals: Some people might think this could be tracked but if you die, all enemies and capsules re-spawn so you can gain infinite animals (unless there's a max such as 999 then maybe they could be tracked?).
Windy Hill:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times
Act 4: Times, Rings
Desert Ruins:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times, Rings
Act 4: Times, Rings
Tropical Coast:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times, Rings
Act 4: Times, Rings
Act ?: Rings
Frozen Factory:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times
Act 4: Times, Rings
Silent Jungle:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times, Rings
Act 4: Times, Rings
Sky Road:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times, Rings
Act 4: Times, Rings
Act ?: Rings
Lava Mountain:
Act 1: Times, Rings
Act 2: Times, Rings
Act 3: Times, Rings
Act 4: Times, Rings
Hidden World:
Act 2: Rings
Act 3: Rings
Act 4: Rings
I shall update this topic if I find anything else out, and of course feel free to add to this list where you see fit.