Inspector Bradstreet is a detective who appears in three short stories: "The Man with the Twisted Lip", "The Adventure of the Blue Carbuncle" and "The Adventure of the Engineer's Thumb". Doyle described him as "a tall, stout official... in a peaked cap and frogged jacket". Sidney Paget's illustrations for the Strand Magazine depict him with a full beard. Beyond this little is revealed about him in the canon.
Bradstreet originally served in Scotland Yard's E (Holborn) Division which associates him with the Bow Street Runners, a forerunner of Scotland Yard. He claims to have been in the force since 1862 ("The Man with the Twisted Lip") but in June 1889 Dr Watson writes he is in B (Chelsea) Division to oversee "The Adventure of the Blue Carbuncle". According to Sherlockian author Jack Tracy, B Division was "one of the twenty-two administrative divisions of the Metropolitan Police Force. Its 5.17 square miles include parts of south Kensington and the south-western section of West-minister [sic?]".
In "The Adventure of the Engineer's Thumb", he accompanied Holmes to Eyford, a village in Berkshire. According to Jack Tracy's The Encyclopaedia Sherlockiana, he was "assigned most likely to the central headquarters staff." Bradstreet is not a martinet; in "The Man with the Twisted Lip" he could have prosecuted the false beggar, but chose to overlook this action to spare Neville St Clair the trauma of shaming his wife and children.