I have a toilet that runs periodically, like maybe half the time after a flush. For what it's worth, it's not a constant run, but rather the water kicks on every 5 minutes or so to top off the tank.
The first thing I checked was the chain. The chain that connects to the flapper is not too short as to leave the flapper unsealed and it not too long as to get tangled up.
So next I removed the flapper and inspected it. No cracks or any signs of wear, really, which is not surprising seeing as the toilet and its innards are all less than four years old. I wiped down the underside of the flapper with a rag and did the same to the O-ring in the tank where the seal is made. Still no dice.
One odd thing I noticed that I've never experienced before with this toilet or others, is that when I am urinating into the toilet after like half way through I very distinctly hear the same noise you hear when you pour a liquid into another liquid (sorry, don't know if there's an official term for that sound). This keeps up for maybe another 2-4 seconds when I stop urinating. Could the increase of liquid in the bowl be overflowing into the tank or some other body of water? Not sure if that oddity would provide any help in diagnosing the problem or not, but thought I'd throw it in there.
My next step is going to be to try replacing the flapper, but I was hoping someone might have some additional insight or ideas into this issue.