Here is a 1920 rogue's gallery of mugshots taken in Calgary, Canada of gentlemen wearing dapper hats. From Vintage Everyday:
After the defeat of the Paris Commune in 1871, the Prefecture of Police of Paris hired a photographer, Eugène Appert, to take portraits of convicted prisoners. In 1888, Alphonse Bertillon invented the modern mug shot featuring full face and profile views, standardizing the lighting and angles. This system was soon adopted throughout Europe, and in the United States and Russia.
(Thanks, Michael-Anne Rauback!)