It's definitely tricky trying to analyze the style of thirteen separate stories. And while each story has a somewhat different style (none of them start the same way, for instance), we're going to focus here on a few common elements that exist in multiple stories, creating a sort of distinctive and recognizable Sherlock Holmes story style. So this might be a bit generalized, but we think it's a good jumping-off point for a closer look at any specific story in this collection.
First off, we have a linear style, which is a way of saying the events of a story are told in chronological order. Unlike Lost, Conan Doyle's stories don't hop around all over the place. Narrator Watson might start us off with a brief mention of the present and how he is remembering a certain case. He also tells his stories in the past tense. But, Watson still tells us the sequence of past events in chronological order; the man doesn't do spoilers in other words. By not giving things away too early, Watson allows readers to experience the mystery as he experienced it back in the day. The mystery stays mysterious in other words. All the characters we encounter tend to tell their stories in chronological order as well, though we'll get back to those narratives in a minute.
First though, it's worth noting Conan Doyle's style on a sentence-level. Linear helps us to describe how Conan Doyle orders events and how the plot develops. On a sentence level, though, we can describe Conan Doyle as pretty effusive, which means wordy and excited. Conan Doyle doesn't always use very long sentences, but he does tend to use lots of adjectives. He makes our narrator Watson downright gushy at times, especially when he's talking about his BFF Holmes or the women he meets.
Seldom have I seen so graceful a figure, so womanly a presence, and so beautiful a face. (Abbey Grange.21)
Watson never just says someone was "gorgeous" or "nice." Why use one adjective when you can use three? This sentence here also demonstrates another aspect of Conan Doyle's style: he's very literary at times, and his language can get pretty flowery. In the above sentence, we also see him making use of some parallel structure and repeating clauses. It's like he has a list of literary terms from an English class and is determined to use them all.
We had sprung to our feet in amazement at this ponderous piece of wreckage, which told us of some sudden and fatal storm far out on the ocean of life. (Priory School.2).
In that sentence alone we have some alliteration ("ponderous piece") and an extended ocean/sailing metaphor. Stylistically, these gushy and literary sections contrast to, and also compliment, the overarching style of these stories: detailed.
Detailed here refers to how much information we get. And we get a lot. We hear details about what people wear, how they look, how the events of a crime occurred (in order, of course), what the weather is like, Holmes's expressions, Watson's feelings, etc. They way detail is presented, however, varies. Conan Doyle's literary and effusive style is one way of presenting detail. But we most often get detail in the form of a newspaper article or a police report.
The minor characters we encounter in each story are always surprisingly well-spoken and long-winded. They always, conveniently, tell us everything we need to know about a particular case. Criminals too always make full confessions, explaining their motives as well as how they did whatever they did. What's interesting is that these characters often sound really similar to each other. But this makes some sense because Watson himself is recalling these stories. We're basically hearing Watson's memories of how these people spoke, which accounts for how similar a lot of these people sound. But Conan Doyle does include certain unique speech traits in a lot of the characters in these stories. We get minor variations on the overall linear, detailed, and sometimes effusive style that tends to dominate all of these stories. Here are two quick examples, the first from the "Norwood Builder" and the second from the "Missing Three Quarter":
"I was very much surprised, therefore, when yesterday, about three o'clock in the afternoon, he walked into my office in the city." (Norwood Builder.37)
"At ten o'clock I went round and saw that all the fellows had gone to roost [....]" (Three-Quarter.20).
Both characters are explaining their problems to Sherlock Holmes with a lot of detail, like the inclusion of time. But while the first character uses "therefore," to make a point, the other character uses slang and funny metaphor to make his point (Norwood Builder.37). We get similar types of details here, but the detail is spoken in different ways.
We also get a lot of graphic detail in the Sherlock Holmes stories. Conan Doyle gives us descriptions of blood and guts at murder scenes that border on the gory at times. Within these stories, we see a lot of contrast between different kinds of detail: romantic versus graphic descriptions, facts versus metaphors. All of these different styles unite in the newspaper clippings that Watson provides for us. In an article entitled "Murder in Westminster" in "The Second Stain," the article opens with a lengthy description of Eduardo Lucas, how well liked he "is" and his fancy house, before moving on to a description of how a detective came into his house and found him murdered (Second Stain.25). Rather than just open with "Eduardo Lucas was murdered," the article uses romantic and later graphic descriptions to build suspense before making the big "reveal."
It's interesting that Conan Doyle's narrative style heavily mimics the style of these newspaper articles. Conan Doyle frequently uses the same sorts of techniques in his own narratives, and the appearance of newspaper articles in these stories helps to demonstrate the journalistic aspects of his style. Or at least the aspects of a rather romanticized journalism.