The Supremes consider two cases that could loosen the standard set by the court in Miranda v. Arizona, which required law officers to inform suspects of their constitutional rights before interrogations. One is debatable; one is a no-brainer.
In the first case, a suspect being read his rights interrupted the officer, declaring "I know my rights." The question is whether a gun that the officer subsequently found when the suspect consented to a search of his home is admissible as evidence. The NY Times claims that precedent cuts against the admissibility of the gun. I have no idea what the case law says, but I just saw Ron Kuby on CNN, and even he seemed to think this was probably permissible.
The no-brainer involves the practice of "two-part" interrogations. The trick works like this: Interrogation one is conducted without Mirandizing the suspect, ideally eliciting a confession. That confession, however, is inadmissible in court. Then, the officers stop and, after a break, start a "new" interrogation, this time Mirandizing the suspect. If they can get her to confirm the confession she made an hour ago, it's now admissible. This one is obviously just a sneaky means of doing an end-run around the procedural requirements of Miranda, and seems unlikely to stand scrutiny.