Here's the Supreme Court's decision on SB1070, the Arizona immigration law. Most of it was struck down, but not a key, controversial element:

It is not clear at this stage and on this record that §2(B), in practice, will require state officers to delay the release of detainees for no reason other than to verify their immigration status.  This would raise constitutional concerns.  And it would disrupt the federal framework to put state officers in the position of holding aliens in custody for possible unlawful presence without federal direction and supervision.  But §2(B) could be read to avoid these concerns.  If the law only requires state officers to conduct a status check during the course of an authorized, lawful detention or after a detainee has been  released, the provision would likely survive preemption—at least absent some showing that it has other consequences that are adverse to federal law and its objectives.  Without the benefit of a definitive interpretation from the state courts, it would be inappropriate to assume §2(B) will be construed in a way that conflicts with federal law.

Translation: Police can still ask to see your proof of U.S. residency.

EDIT: But as Think Progress notes, this decision leaves room for this portion of the case to be decided more definitively later. If there is evidence that this is being used for racial profiling, it’s possible it could get struck down later.