Polishing was the act of giving an object a smooth, lustrous surface. This could be accomplished using a special polish, or it could simply involve buffing the object in question with a cloth.
In 1930, Edith Keeler praised James T. Kirk and Spock's work in the basement of the Twenty-First Street Mission, saying it looked as though it had been scrubbed and polished. (TOS: "The City on the Edge of Forever")
In 2152, Trip Tucker sarcastically offered to polish all the handles and rails in Enterprise NX-01's Engineering while the ship unexpectedly changed course for the Pernaia system. After Jonathan Archer gave him a look, he decided to purge the impulse reactors instead. (ENT: "The Seventh")
In 2268, when Cyrano Jones offered to sell the bartender on Deep Space Station K-7 Antarean glow water, he replied that he used it to polish the multiple Spican flame gems he'd already sold him. (TOS: "The Trouble with Tribbles")
In 2369, Quark berated Rom for returning a patron's full currency pouch, giving him the punishment of polishing every rail in Quark's. Soon after, spying a lazy Nog, Rom passed on the punishment to him, saying he wanted every rail to sparkle. (DS9: "The Nagus")
In 2373, in The Doctor's original version of Doctor's Family Program Beta-Rho,"Kenneth"'s holographic wife, Charlene, stood by the door and asked Belle and Jeffrey if their shoes were polished. (VOY: "Real Life")
In 2375, as an ensign, Nog polished Benjamin Sisko's desk, much to Sisko's delight. Nog explained that he did so because he thought it was looking a little dull. In reality, he had loaned the desk to Al Lorenzo temporarily so the man could take a holophoto of himself sitting behind it, and had just returned it to the office when Sisko discovered him. (DS9: "Treachery, Faith and the Great River")
In 2377, assuring Irina that Harry Kim, her new co-pilot in the Antarian Trans-stellar Rally was a good choice despite the fact that her ship was damaged, B'Elanna Torres remarked, "He'll not only fix your ship, he'll polish the bulkheads. " (VOY: "Drive")