Letter of the law is that you must hand in any found property - Posting “Is this yours?” on Facebook doesn’t cut it. Failure to hand in to Police can potentially see you done for “Theft by Finding”.
I’ve only had two or three rings claimed from Lost & Found, and they were surface-find (ie recent) fancies with inscriptions.
It’s not always the owners who get them either, sometimes it’s the Insurance Companies if a claim has been paid out.
Plain bands (ie with no names, or unique designs) always come back to me, as do ‘green’ golds (Crusty 9ct), silvers and deep/old finds.
Even when a ring is returned, legally you still don’t own it (Although I have never heard of any issues with onselling in this case, and you have the Police slip to show you have made the appropriate effort) If you tried to sell on TM, and someone can PROVE it’s theirs then you are required to surrender it - Best way would be to resurrender it to the Police and give the ‘claimant’ the Lost Property receipt number - If it’s theirs then the Police will make sure, if not, the person won’t follow up and three months later you get it back.
I had one returned to me as unclaimed, despite having a name and date inside it. Bit of legwork traced it to a lady in Argentina, who luckily blogged about losing it when she was over here on holiday - date was her wedding day.
Some people, probably a majority to be truthful, just pocket them.
Golden (haha) Rule: What goes in the pouch, stays in the pouch - Don’t be tempted to show rings off.
There’s a scam in the US where a hunter shows a ring to pretty girl, the girl then reports it as ‘found, but he won’t give it back’ to Police…and they have a good description of the ring to prove it! Copper turns up and you are forced to hand it over.
Follow your own moral compass, and good hunting!