Oh cool topic! I will do it too since I actually *gasp* have records.
Disclaimer: I can and will be proud of Sonic06 records.
BEST
Wave Ocean, Sonic, 100000 points - Read it and weep. 100,000 points. That's all there is to it, really... that's just sweet.
Kingdom Valley Hard, Sonic, 529 rings - Over 200 rings less than my Kingdom Valley Normal stat, but still a record by a wide margin in such a long and treacherous stage - to hold those rings through the hard mode versions of Silver's area and the highspeed section.
Radical Train Hard, Sonic, 478 rings - I am proud of this one because I even remember how intense it was to get all the rings I did in the first area before the train crashes. And not only that, I managed to keep them all through the highspeed section and ended with only 15 rings fewer than I did in my normal Radical Train stat.
Dusty Desert Hard, Silver, 68970 - I spent a bit of time in this stage and made it one of my niches... but now this is the only record of mine that remains in all Dusty Desert Silver entries, so yea I'm proud of this and it stood the test of time. Comment indicates that I got the trick shot on my first attempt (perhaps after destroying the vases first) and that I was man enough to avoid the golems as Amy instead of being lured in by the false promise of an extra 2000 points. That's pretty cool.
Flame Core Hard, Sonic, 83620 - A record by a wiide margin, and not a placeholder but an ideal run, without getting hit and my balls were so large that I went ahead and defeated 2 golems in the Knuckles area.
Radical Train, Sonic, 105580 - I remember watching with great interest and awe at RPG's Radical Train score run, only to later begin to realize that there was nothing particularly insane about his run, neither was there anything I couldn't do, either. I ended up beating his score, apparently doing it slower than his, but making up for it by getting a ton more rings than he did, to beat his score by a small margin... this is mostly thanks to handling the highspeed section much better. RPG has since never gotten around to taking back this stage.
Mission 6, Shadow, 43110 - I believe this is the record I attained by getting confused between the town missions and writing down the incorrect record to beat. I ended getting a score well over the current record (back then anyway) .. which I got merely by trying to do this run getting everything and without making any mistakes. I don't know if some of these are my best records, but I'm definitely proud of them, and the fact remains that they are still not beaten.
WORST
Babylon Garden, 2:31:15 - I don't remember what exactly went wrong here, but going by my comment, ("I made 50 mistakes") it must have been a lot more than just the last jump. The last jump is what I remember most specifically however... it's definitely possible to get an X-rank trick on that ramp, but I could not pull it off with any consistency and apparently have no idea what needs to be done specifically to keep it from failing. The last jump is what kept me from bothering to perfect the rest of the run, but I find it quite unsettling that this is my ONLY time attack record in the game that hasn't been beaten yet. (to boot, my only Race record is also for Babylon Garden... ew)
Wave Ocean, Silver, 73880 - As my comment on the record indicates, I lost a score bonus for killing the invisible crabs properly. Considering that this is a short stage and I have the record by such a small margin, this can be taken relatively easily. Unless people actually don't wait for a score bonus at the very start of the stage...
Kingdom Valley, Sonic, 125450 - My time bonus was not even 4000, meaning that anyone who doesn't hurry can go and defeat everything and get all the rings before the highspeed section to get a much higher, "slow" score much like Aquatic Base Sonic. (even though nobody did and my record still stands by such a large margin)
and finally: very many much of my Sonic06 records aren't groundbreaking or are easily beatable but still haven't yet, quite apparently because simply nobody bothered to try, at least not with enough serious effort.