I've often heard it said that Sherlock Holmes — the deerstalker's classic model — would never have worn one. Likewise, although at least one of the few reliable witnesses identified Jack the Ripper as sporting a deerstalker, it's been suggested by a few researchers (Gian Quasar's Scarlet Autumn comes to mind) that such headgear would have made the Ripper stick out like a sore thumb, that it would have been laughably incongruous in 1888 Whitechapel.
How much truth is there to this? Would it have been an absurd fashion choice outside of actual deer stalking?
Deerstalker hats were most popular during the late 19th century especially from around 1870 to 1890. They were considered stereotypical upper-middle-class outdoorswear, along with Norfolk jackets and knickerbockers. As the name suggests "deerstalker caps, also called "fore-and-aft caps", were originally used for hunting and other outdoor pursuits. They would not have been normally worn in urban areas, or indoors, as Holmes is sometimes pictured doing. A man of Holme's class (marginally middle-class professional?) would more normally have worn a top hat or possibly a bowler hat, depending on the weather and occasion. Top hats in the 1880s were shorter and had a more "waisted" profile than the more cylindrical "stovepipe" style that had been popular in the 1860s.
More speculativley, I've seen two major fan theories over the years as to why Sherlock Holmes is often depicted with a deerstalker cap. The first is that it was done to emphasize to Victorian readers his eccentric nature -- wearing a deerstalking inside his city flat! The other more prosaic hypothesis is that the illustrator for the early Sherlock stories, Sidney Paget, was himself rather fond of a deerstalker cap for wearing on rainy days.