His popularity is probably just as popular today as it was a century or so ago.
The Sherlock Holmes stories are structured in a very set way, most of the books are told in this style. They begin with a narration by Watson, the tale is often begun with a sentence saying how "particularly interesting" this case was compared to all the others he has been involved with, then we find ourselves in Sherlock's house. When we are situated here there is usually a visitor who turns out to be the victim of the story. Holmes introduces the visitor to Watson and us and then begins to inquire into why they are here. As they begin to explain who they are and where they came from Holmes says something that almost "shows off" his intelligence and quite takes aback the visitor and even Watson. It is to show how clever he is and how simple all the logic and deduction is once you have heard Holmes explain how it works.
""You must have started early, and yet you had a good drive in a dogcart, along heavy roads, before you reached the station.".
The lady gave a violent start and stared in bewilderment at my companion.
"There is no mystery, my dear madam" said he, smiling. The left arm of your jacket is spattered with mud in no less than seven places. The marks are perfectly fresh. There is no vehicle save a dogcart which throws up mud in that way, and then only when you sit on the left hand side of the driver."".
We then are told the mystery that Holmes is wanted to solve by the visitor and every so often Holmes will interrupt and ask a seemingly pointless question which turns out to be a big clue in solving it.
""One horse?", interjected Holmes. .
"Yes, only one".
"Did you observe the colour?".
"Yes, I saw it by the sidelights when I was stepping into the carriage. It was a chestnut.".
"Tired looking or fresh?".
"Oh, fresh and glossy"".
From here we usually have Holmes leave his home along with Watson to investigate the scene of the crime or place of mystery.