This question is starting to gain more attention again after the release of Netflix's "Enola Holmes" as there are claims of the film infringing on the copyright belonging to Arthur Conan Doyle’s family.
Sir Arthur Conan Doyle created the character Sherlock Holmes in 1887, so he is the original owner of Sherlock Holmes. Sir Arthur Conan Doyle however, died in 1930.
Many people would believe that the case is simple, that a writer's work is copyrighted or it isn't? However, this isn't the case. In the UK, copyright lasts for 70 years after an author’s death; Conan Doyle died in 1930, so the copyright has expired. Having said that, in the USA some copyrights extend for 95 years from the date of the work’s first publication.
Consequently, out of a total of 56 short stories and four novels to feature Holmes, a final 10 short stories that Conan Doyle published between 1923 and 1927 are still in copyright to The Conan Doyle Estate.
The Conan Doyle Estate are distant descendants of Conan Doyle and other people. The Doyle Estate claim ownership of certain intellectual property rights to Conan Doyle's works and grants licenses to use these rights.
The Doyle Estate made claims against Miramax’s 2015 release Mr. Holmes claiming that it was infringing on the handful of copyrights that The Doyle Estate still holds. The Doyle Estate wants to control the character so they can make a ton of money.
In the 10 copyrighted stories that they own, the estate observes a Holmes who is more empathetic and given to expressing emotion, so they argue that these characteristics are copyrighted to them.
The Doyle Estate is now targeting the new movie Enola Holmes as they argue that Netflix has given Sherlock legally actionable (copyrighted) feelings. In "Enola", Sherlock being worried about his sister Enola is an emotion similar to what Doyle only introduced in his later stories, making these emotions trademarked elements still controlled by the family.
This is money motivated. The Holmes stories written before 1923 were released into the public domain back in 2014, should this be the case for the remaining 10 owned by The Doyle Estate? Creative writers that want their chance to tell a story within his world would argue yes.