If you have permanent resident status, there is a requirement to have a green card in one's "possession" at all times when in the US. Granted it's not a high priority for the federal government to enforce. I'm not even sure what that means as a practical matter, such as walking on the beach or going swimming.
http://www.uscis.gov/green-card/after-green-card-granted
A green card is issued to all permanent residents as proof that they are authorized to live and work in the United States. If you are a permanent resident age 18 or older, you are required to have a valid green card in your possession at all times.
That may be the only requirement in US law that a member of the public must carry identification when out and about. Nobody else is required to carry ID, unlike some parts of the world where there is a standard government-issued ID. Several of my high school teachers were from Europe, and they noted that even as children they had to carry their national identity papers (which are now cards). I've read that national identity cards are now the most common way for people to travel across borders in the EU, although there are a few EU countries that don't issue them. The UK apparently discontinued theirs.