Much of the pleasure we derive from reading Arthur Conan Doyle's stories about Sherlock Holmes comes from enjoying Holmes' extraordinary displays of powers of observation and deduction. In the story "The Red-Headed League," we encounter these powers early on when Holmes remarks to Watson:
Beyond the obvious facts that he has at some time done manual labour, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else.
Of course these facts are not obvious to Watson, and in his explanation of how he arrived at his conclusions, Holmes establishes his expertise and reveals his methods. The first skill he uses is one of close observation of details (in many cases with his iconic magnifying glass). The second is having built up a storehouse of facts (such as knowledge of tattoos and tobacco products) which allow him to deduce facts about people and evidence that the less well-informed might miss. Finally, he uses logic, unprejudiced by conventional preconceptions.
One characteristic of Holmes' method is that it is based on analysis, on taking time to think through a problem and formulate matters to be investigated, rather than simply leaping in and starting to investigate at random. After Holmes takes on the case we read the following bit of dialogue:
"What are you going to do, then?" I [Watson] asked.
"To smoke," he [Holmes] answered. "It is quite a three-pipe problem, and I beg that you won't speak to me for fifty minutes."
Once Holmes has worked out the problem in his mind, he decides to investigate in person. The focus of his investigation is elimination of wrong answers until the only one left is the correct one, or as is famously said in The Sign of the Four, "when you have eliminated the impossible, whatever remains, however improbable, must be the truth." The final stage in the crime-solving process is apprehension of the criminal.