But before you decide to exercise that right to remain silent you better tell the police that that is exactly what you are doing.
The Supreme Court ruled today in Berghuis v. Thompkins that a criminal suspect must specifically invoke the right against self-incrimination in order for constitutional protections to apply.
From SCOTUSblog’s analysis of the decision
if a suspect does not want to talk to police — that is, to invoke a right to silence — he must say so, with a clear statement because it is not enough to sit silently or to remain uncooperative, even through a long session; and, second, if the suspect finally answers a suggestive question with a one-word response that amounts to a confession, that, by itself, will be understood as a waiver of the right to silence and the statement can be used as evidence. Police need not obtain an explicit waiver of that right.
In this case, that one word, after several hours of questioning met with silence, was enough to sentence the defendant to life.
If you are arrested now, your silence while being questioned by police does not amount to an invocation of his Miranda right to remain silent. Previously (and until many PDs update their interrogation policies) the police were generally required to get you to explicitly waive your right (usually by signing a document to that effect). Now they get to assume you waived it unless you explicitly invoke it.