US citizens are required by law to carry ID. Be it a drivers license, state ID, or passport.
My understanding has been that that is not the case, in general, and that American citizens are perfectly welcome to walk around, ride as passengers in private automobiles, and ride as passengers in private airplanes without carrying any ID whatsoever. (Ironically, the only time I remember showing a police officer my driver's license, I was a pedestrian.)
Drivers and pilots are required to have licenses with them.
Airlines and Amtrak typically want to see identification from many or all of their passengers.
Commuter railroads and local transit systems generally don't care who you are.
John Gilmore is one person who has spent a lot of time on this issue. I believe there is also someone at Harvard Law School who has determined emperically that it actually is possible to fly on a commercial flight while refusing to show ID.
A driver's license does not prove citizenship, and I believe that when employers are verifying that an employee is eligible to work in the US, they can use a driver's license to verify that the person in the photograph on the driver's license is the person with the name printed on the driver's license, but then another document is needed to establish that the person with that name is a US citizen.