(covers information from several alternate timelines)
Before Spock embarked on a scientific mission into the space amoeba in 2268, Leonard McCoy expressed his lack of understanding of Vulcan dignity, so Spock asked him to employ a human superstition and wish him luck. (TOS: "The Immunity Syndrome")
Julian Bashir and Miles O'Brien wished Odo "good luck" before the constable departed to deal with Changeling infiltration on Earth. Odo appreciated the sentiment, though he said that he did not personally believe in luck. (DS9: "Homefront")
Not long after this, when Odo and Major Kira Nerys were having a discussion regarding Quark's being disturbed by the noise made by Odo's shapeshifting, Kira commented that it was just Quark's luck that Odo would be assigned quarters right above his and Odo replied that luck had nothing to do with it, much to Kira's amusement. Later, when Bajoran First Minister Shakaar Edon was on Deep Space 9, Kira wished him luck prior to his giving a speech. Sometime after this, Worf noted to Odo that Miles O'Brien had a tendency to drop by his quarters announced and suggested that O'Brien might stop if he was less hospitable. Despite his lack of belief in it, Odo wished Worf "good luck." (DS9: "Crossfire")
The Klingon position on luck has not been explicitly stated; however, the fact that Klingonese uses "qapla'" ("success") in place of the human "good luck", and Ch'Pok's comment that "fate is a human concept", would suggest that they do not believe in it.
When Seven of Nine discussed a precaution against Theta radiation with Tuvok, he stated that "With any luck, we won't be needing it." Seven replied that she thought that Vulcans didn't believe in luck, but Tuvok replied that serving with Kathryn Janeway had taught him otherwise. Seven agreed that she did seem to succeed more often than random chance would allow for, and stated that she would factor it into her calculations. (VOY: "Juggernaut")
"Beginner's luck" was a type of luck sometimes attributed to those who win a game on their first try, sometimes even against an experienced opponent. In 2378, when Icheb won a game of kal-toh against Tuvok, he attributed to beginner's luck, though Tuvok recognized his loss as being due to lowered neuropeptide levels caused by a Vulcan neurodegenerative disease. (VOY: "Endgame")
A bit later that year, Admiral Kathryn Janeway arrived from the future with a plan to return the USS Voyager to the Alpha Quadrant by utilizing a Borg transwarp hub. Her Captain counterpart was wary, stating that they shouldn't push their luck, but the admiral replied that luck wouldn't have anything to do with it, as she had developed weapons and technology to defeat the Borg. (VOY: "Endgame")