What methods did Sherlock Holmes use to solve mysteries?

Early in his career Sherlock Holmes trained himself to become a "private consulting detective." He told Watson when they first started sharing rooms at Baker Street that he was not interested in any knowledge that was not of use to him in his work. Their conversation on the subject is contained in the opening chapter of the novel A Study in Scarlet (1887). Holmes uses his extraordinary powers of deduction to solve mysteries, but he also relies heavily on his magnifying glass, microscope, and chemistry apparatus. He has an expert knowledge of all sorts of scientific subjects that are useful in detection. He is also a a great reader of anything that may be of use to him now or in the future. He reads all the newspapers and keeps clippings of articles about crimes and criminals. He also has his own personal set of files, so that he is better informed on the subject of crime that almost anyone in Europe. The police frequently come to him for advice. This is advantageous because the police are usually willing to give him help in return if he asks for it. Using the police for information and assistance is another of the methods Holmes employs to solve mysteries.

In "The Red-Headed League," Sherlock Holmes becomes interested in Jabez Wilson's apparently petty problem because he suspects that Wilson's assistant, who calls himself Vincent Spaulding, is really the notorious John Clay, described later by a man from Scotland Yard as:

“John Clay, the murderer, thief, smasher, and forger."

Holmes knows all about Clay's career because he is so well informed about crime and criminals through his reading, record-keeping, and many informants. He deduces that Clay must have some important ulterior motive for working in Wilson's shop. Holmes' cases almost always take him to the scene of the crime—or, in this case, to the scene of the anticipated crime. This is the author's way of bringing the reader, in imagination, to that same scene. One of Holmes' most commonly employed methods of solving mysteries is to make minute examinations of crime scenes, where he picks up minute clues that could easily escape the attention of police professionals. From what Wilson tells him about Clay's supposed interest in photography, Holmes makes deductions that would never occur to the "not over-bright pawnbroker."

“Oh, he has his faults, too,” said Mr. Wilson. “Never was such a fellow for photography. Snapping away with a camera when he ought to be improving his mind, and then diving down into the cellar like a rabbit into its hole to develop his pictures."

Holmes quickly deduces that Clay must be digging a tunnel, and by inspecting the neighborhood he deduces that the tunnel must be aimed at the Coburg branch of the City and Suburban Bank, where he later learns they are storing an enormous fortune in French gold coins.

So Holmes relies on his accumulated knowledge, his record-keeping, his personal investigation, and his deductive powers to capture John Clay and send him to the gallows. It is fortunate for the directors of the bank that Jabez Wilson came to Holmes with his trivial problem involving the nonexistent and now defunct League of Red-Headed Men.