File a report in the paper that a ring was found, or call the police and tell them you know of a ring that was found and not turned in. See what happens.
If you find money on the ground, small denominations, I don't think there is a problem with keeping it--unless you saw it fall you'll never know where it came from. But larger amounts (I won't try to define small vs large) should entail a trip to the police station. They are supposed to return the found item to the finder if it isn't claimed.
If you ride the bus every day and find money at the bus stop, a general "did you lose anything yesterday" inquiry should happen, whether asking fellow riders or posting a note at the bus stop. There are lots of people who can't afford to lose even small amounts of cash or personal items.