The only stats I keep are the number of putts on each hole, and the distance of the first putt. I started doing that when I was a teenager because my dad did after reading an article about it in Golf Digest. Sam Snead had a system he used to score his putting. Maybe once a season I'll actually bother to calculate my putting score, if I feel like I had a particularly great or particularly horrid day on the greens.
The only other thing I note is penalty strokes, I'll add a B for OB, L for lost ball, W for water hazard, H for lateral hazard and U for unplayable (I'm perversely proud of the fact that I once had a round with all five marks and still broke 80)
However, I do absolutely nothing with this information I collect. I don't put it into a spreadsheet, or look at it later to try to improve my game. I just record it out of force of habit, because I started doing it many years ago because my dad did it. I still have the scorecard for almost every round I ever played from my very first one, because my dad saved all his cards. I have several shoeboxes full now (if I played as often as some of you guys do, I'd need a full sized filing cabinet to hold them all!)
My dad always said he saved them because someday when he was too old to play he could look back at them and recall some of his best rounds. He's still playing, so hasn't tested that theory yet. I don't know if I'll ever look back them, but having saved them since my first round at age 12 (when I shot an 82 for nine holes, so at least I can demonstrate an improving trend) it would be a shame to toss them now so I guess I'm stuck with the collection!