While you may think you're ok to travel domestically within the United States freely, you may not realize that you should technically be carrying your immigration documents at all times to avoid issues. Especially in recent tense times surrounding immigration issues, it's important you know whether you have to carry your documents!
Whether crossing state borders or going on a holiday out of town, you are generally recommended to always be carrying your immigration documents with you in order to avoid problems with local, state, and federal authorities.
If you are 18 or older, technically yes, you do have to carry some form of your official registration documents with you.
For Green Card Holders (Permanent Residents)
Section 264(e) of the Immigration and Nationality Act (I.N.A.) requires all residents to have “at all times” official evidence of LPR status:
(e) Personal possession of registration or receipt card; penalties
Every alien, eighteen years of age and over, shall at all times carry with him and have in his personal possession any certificate of alien registration or alien registration receipt card issued to him pursuant to subsection (d). Any alien who fails to comply with the provisions of this subsection shall be guilty of a misdemeanor and shall upon conviction for each offense be fined not to exceed $100 or be imprisoned not more than thirty days, or both.
It's really as simple as that. If you're found to not be carrying your green card you can get a fine or be imprisoned!
What official documents should nonimmigrants carry?
For nonimmigrants though, it's a bit more confusing.
Technically you should have your evidence of your I-94, which is your proof of legal admission and status in the country. It used to be a small piece of paper but this isn't the case anymore as it's all done digitally. If you were given a paper I-94 be sure to carry that with you!
Your passport does contain a stamp in most cases (which is somewhat like a physical version of the digital I-94), but carrying your passport at all times can be more dangerous than it's worth.
Should I really carry my passport?
It's difficult because if you are traveling and you can't prove your residency, identity, and official status in the United States, you may become unstuck.
While I don't like to make any recommendations, because I am not an attorney, I prefer to personally carry copies to show that I could make the real documents available in a short time.
I carry a copy of my passport's photo page, a copy of my visa, and a digital print out of my I-94.
This is not advice for you though, you do whatever makes you feel comfortable and you should always consult a legal professional.
Important note for international travelers
If you're traveling outside the U.S. you absolutely MUST have all your documents on you, and if you are flying domestically, your passport will work with full effect in all states (which sometimes other forms of ID do not).