Tara Fairchild
Yes, it has been used in reference to the police like Jim said New York’s finest, or LA‘s finest. Not sure where it came from or why it exists, but you don’t hear it too often these days. I would even go so far as to say when it is used in the rarity these days it’s more with sarcasm. 
Jim,man of GOD, friend to GOD
Yes. I have definitely heard that term used to refer to the police life New York’s finest Chicago’s finest. I’m not sure that it’s appropriate these days, but I have heard it.
Harmony Finder
Hi Brian, i was curious, so i asked the echo the definition as well, and got exactly the same answer. :)
Brandon Cross
Déjà Brew Diane
King Tripple Whopper
Brian williams
Yes, I understand finest means the best. But I've never heard the police referred to as finest either here or anything from the states I've heard. Just wondered whether it was a more common term than I realised. 
Leftist Point of View
the term “finest” as a nickname for the police is actually pretty old-school and mainly comes from New York City. The NYPD is often called “New York’s Finest.” Over time, that got shortened to “the finest” as a kind of nickname for the police in a more general sense. So, if you hear someone say it that way, they’re probably just borrowing that old NYC slang.
Déjà Brew Diane
Nicole a.k.a. the Brassknucklebeauty
King Tripple Whopper
Finest means of the highest quality! It means that those mince pies have the best ingredients out of all others!