It seems that nobody at Dead Guy has written about the first mystery, so I’ll volunteer. After all, I’m a librarian, with scads of resources at my fingertips. And it turns out one of those resources is super cool: Colorado College Special Collections has a facsimile edition of the manuscript of Edgar Allan Poe’s “The Murders in the Rue Morgue,” a short story published in Graham’s Magazine in 1841. Its protagonist, Inspector Dupin, is the first fictional detective in English literature, maybe any literature.
As you can see from this image of the first page of the manuscript, Poe at one point titled the story “The Murders in the Rue Trianon-Bas.” The facsimile is available at many libraries; Open Worldcat will find the one closest to you. Or you can read a digital version of the entire facsimile at the Internet Archive. If Poe’s handwriting is too difficult for you to decipher, you can try this transcription instead.
We are lucky the manuscript survives at all. According to the Edgar Allan Poe Society of Baltimore, “J. M. Johnston was an apprentice at the firm of Barrett and Thrasher, which was at No. 33 Carter's Alley in Philadelphia. This firm was responsible for typesetting material for Graham's Magazine, in which the story first appeared. According to Johnston, 'the revised proof [was] read in the Saturday Evening Post Office,' the Post and Graham's Magazine both being owned by George R. Graham. As was typical of the process, the manuscript was thrown in a wastebasket after the text had been set in type. Johnston asked for permission to retrieve the handwritten pages, and he kept them in his possession until July 1881, when he appears to have sold the manuscript to George W. Childs. When Childs died in 1894, the manuscript was left, with much other literary material, to the Drexel Institute, where it remained for several decades. A facsimile was printed by George Barrie in 1896. On October 17-18, 1944, a large portion of the Drexel collection, including Poe's manuscript, was sold at auction. It was purchased by Richard Gimbel for $34,000. Gimbel donated his collection to the Free Library of Philadelphia in 1974.”
If you’d like to see that original manuscript, you'll have to visit the Free Library.