The scientific method was a set of techniques used in various sciences in correcting, integrating, and acquiring knowledge. The basic tenets of the method was to observe a phenomenon, theorize on the cause, and then attempt to prove the theory.
In 2364, Beverly Crusher used the basic scientific method to determine that the microbrain was a living, yet inorganic creature. (TNG: "Home Soil")
In 2369, while trapped in a game of Chula after a fraught first contact with the Wadi, Bashir claimed to have solved a puzzle using the scientific method, but then immediately admitted that he had in fact just guessed. (DS9: "Move Along Home")
In 2370, when Constable Odo impressed Doctor Mora Pol with his investigative skills, he commented that he was beginning to think that the scientific method and the police method had a lot in common, Odo agreed. (DS9: "The Alternate")