The "public safety exception" allows law enforcement to question a subject, and the prosecution to use his answers against him at trial, even if he wasn't read his rights. To be clear, it's not that someone posing a threat to public safety doesn't have constitutional rights; it's that the threat to public safety has been determined to trump them [source: Benoit].
The exemption was established in 1984, when the Supreme Court heard the appeal of New York v. Quarles. Benjamin Quarles was in the custody of police in a grocery store in 1980, identified by a rape victim as the man who'd assaulted her. The officer frisked him for weapons and found an empty gun holster. The officer asked Quarles where the gun was, and Quarles told him he'd hid it "over there," pointing to an empty milk container. The officer found it and secured it. He then arrested the man and read him his Miranda rights.
The gun evidence was thrown out at trial under Miranda. An appellate court agreed it was inadmissible. The Supreme Court did not.
The Supreme Court found that locating the missing gun was critical to securing public safety, and questions prompted by a reasonable concern for public safety were exempt from the Miranda requirement. Quarles' answer to the officer's question and the gun it led to were therefore admissible in court [source: Benoit].
In April 2013, the FBI invoked this exemption in questioning a suspect in the Boston Marathon bombing. During their "urgent public safety interview" with Dzhokhar Tsarnaev in his hospital room, he confessed to the bombing and answered questions regarding any other attacks still in the works. His request for a lawyer was denied several times. After two days of questioning, Tsarnaev was read his Miranda rights on a judge's orders and went silent. If a court determines the interrogation meets the public safety exception, the statements Tsarnaev made before he was read his rights may be used against him at trial [source: Serrano].
The same exception was invoked in the 2010 questioning of Faisal Shahzad, who failed in his attempt to bomb New York City's Times Square. But Shahzad, after being Mirandized, kept talking -- and never stopped [source: Johnson].
He pleaded guilty at trial, warned Americans to brace themselves for more attacks, and was sentenced to life in prison [source: Katersky]. The public safety exception never came up.