Per U.S. law you are entitled to counsel with an attorney and the government must provide you one if you cannot afford one. Whatever you do, do not say a thing until you are provided with an attorney.
Additionally, the U.S. is a signatory to treaties which require that the appropriate foreign mission must be contacted within 72 hours of detention of a foreign national by law enforcement (even if they are countries the U.S. doesn't have relationships with, like North Korea or Iran, those nations still have a "Protecting Power" in the United States to act in their stead. Protecting power is a any nation acting in the interests of a requesting nation to a nation that the requesting nation does not have formal diplomatic relations with. In the case of those two powers, North Korea has no protecting power in the United States, however Sweden acts as the protecting power for the U.S. in North Korea. For Iran, Pakistan represents them in the United States). Suffice to say, if they think you're a foreign person, they have three days to figure out which country and call there consulate or else it's going to be an international incident (if they catch you in Michigan they'll likely call the Canadians just to be sure, so they'll know.). The foreign mission will be permitted to do a wellness check and discuss legal matters with you and this is treated as privileged and confidential as if you were meeting with an attorney (don't be surprised if they are an attorney in U.S. Law as well. Part of the Embassy's role is to know the laws of the host country in case one of their citizens gets into trouble.).
As to what you carry, any form of State or Federal Government issued I.D. with a picture as part of the I.D. will suffice (this is why your passport is valid but your birth certificate and Social Security card are not, as they are issued without picture I.D. and usually you're quite young on receipt). States issue I.D. cards similar to drivers licenses for non-driving persons with all elements of the license required for identification purposes and they are valid in all U.S. territorial holdings (not just the 50 states, but U.S. territories as well) and typically are identifiable in both Canada and Mexico. In addition to the passport and licenses, if you are a member of the U.S. military or civilian D.O.D. employee or are a dependent family member of either, you should be issued a Military ID card, which is also valid.
Also for all driver's liscenses, a valid learner's permit will also suffice for ID purposes.
This may not be done in ideal time near the U.S.-Mexico border as timely but this due to the ongoing migrant crisis. Canada is much more quiet by illegal border crossing numbers, though it gets it's show.
As a final fun fact: The U.S.-Canada border is the longest border between two countries anywhere in the world. So if you have the misfortune to be a hundred miles inland from the border and they think you crossed it, you have worse luck then a shark attack victim.