As mentioned in another answer, in the US, Miranda rights are attached to interrogations, not arrests.
One crewmember of the Enterprise-D was interrogated without being told of his right to remain silent, though he was offered a lawyer:
PICARD: Please sit down, Mister Tarses. For the record, will you tell us your name and position?
TARSES: Simon Tarses, Crewman First Class, medical technician.
PICARD: I assure you this is an informal inquiry. We are not accusing you of anything. However, if you would like counsel, it can be provided for you.
TARSES: No, sir. I have nothing to hide.
Later, he invoked the right to remain silent on advice of counsel (again, he had not been previously advised of this right, so Riker had to tell him about it):
PICARD: This hearing is convened on Stardate 44780 as a continuing inquiry into the activities of Crewman Simon Tarses. Mister Tarses, for your own protection, I have assigned a counsel to you in the person of Commander William Riker.
SABIN: Isn't it true that the paternal grandfather of whom you speak was not a Vulcan but was in fact a Romulan? That it is Romulan blood you carry and a Romulan heritage that you honour?
(Riker whispers in Simon's ear)
SABIN: We're waiting, Mister Tarses.
TARSES: On the advice of my counsel I refuse to answer that question, in that the answer may serve to incriminate me.
Later, we learn that Worf is aghast that he did this, but Picard defends it by calling it "the Seventh Guarantee" and also mentions a "Constitution" of some sort:
WORF: He refused to answer the question about his Romulan grandfather.
PICARD: That is not a crime, Worf. Nor can we infer his guilt because he didn't respond.
WORF: Sir, if a man were not afraid of the truth, he would answer.
PICARD: Oh, no. We cannot allow ourselves think that. The Seventh Guarantee is one of the most important rights granted by the Federation. We cannot take a fundamental principle of the Constitution and turn it against a citizen.
In conclusion: The right to remain silent clearly exists, but it appears that Starfleet personnel do not need to be advised of it before being interrogated in a Starfleet setting. The right to an attorney also exists, and it appears that personnel are routinely advised of that right (whether this is mandatory is less clear). We do not know whether these rights also apply to civilians. However, it seems unlikely that civilians would be accorded fewer rights than Starfleet personnel, in my entirely subjective opinion.
All quotes from Star Trek: The Next Generation episode 4x21 "The Drumhead".