Under what rule do police have to tell suspects their rights?

Under the Police and Criminal Evidence Act, 1984, an arresting officer in the UK has to caution the suspect that anything they say may be taken as evidence and used against them in the court of law. This means they have the right to remain silent and not incriminate themselves. However under certain circumstances (such as an extremely drunk or violent suspect) this caution does not have to be given straight away, but must be done so before the detainee is interviewed.

Added: (in the US) In conformance with the rulings emanating from the US Supreme Court ruling in 'Miranda v. Arizona,' when the questioning of a suspect not in custody passes from the 'fact-finding' stage and reaches the 'accusatory' stage, it is required.

For an actual arrestee - although they are usuallyinformed of their "Miranda rights" as soon as is practicable after they are placed under arrest, it is not required by law until the custodial questioning about their alleged offense actually begins.

It is not unusual for an arrestee to be 'advised' several times (i.e.: at the time of their arrest, usually by the arresting officer), as well as once again (by investigators) prior to the beginning of their actual questioning.