The sample case used in this guide is Jourdain vs. Patton, which took place in 1818. Its docket number at the Louisiana Supreme Court is 304. In the old series of Martin's Louisiana Term Report the case is found in volume 5, page 615 (abbreviated to: 5 Mart (o.s.) 615 (1818). The court report can be found in both Nexis Uni and The Caselaw Access Project.
Jourdain vs. Patton is also among the digitized items in the Historical Archives of the Supreme Court of Louisiana and can be retrieved from the repository. Most cases are in handwriting, meaning you will need to transcribe them into a digitized text.
Translation: “To the Honorable James Pitot Judge of the Parish Court for the Parish and City of New Orleans. The Petition of Elisabeth Godman, Widow of Baptist Jordain dec.[eased] of the City of N. Orleans
That on the ninth day of Jannuary last a certain negro man slave named James belonging to the Widow of Charles Patton also of the City of N. Orleans did put out the only good eye of your Petitioner’s negro man slave named Mange, not only disabling by that mean of being of any use but rendering him a burden to your Petitioner, that the said negro Mange was before and at the time of being made blind [as afore said], a valuable servant not only...”
Researchers should familiarize themselves with precise legal vocabulary used by the court. An excellent source for checking the terms you wish to search is any edition of William K. Dart and Edward F. White's Louisiana Digest Annotated (1917; Second series, 1937; new 1951; the latter is in our Reference Collection KFL 57.L66), which gives case citations as well as cross-references to other useful terms. Another useful source is West's Louisiana Digest 1809 to Date (Reference Collection KFL 57.L66).
Another particularly useful tool for finding case files that deal with portions of the state's constitution or legislative acts is Theodore Roehl's Annotations to the Statute Law of the State of Louisiana (New Orleans: F. F. Hansell & Bro., Ltd., 1917), a copy of which is in the Dart Collection (Mss 140). This breaks down the Civil Code, Code of Practice, Constitutions (1812-1913), and Acts of the Legislature (1822-1916) into their articles, sections, and acts, and lists the legal citations for all Supreme Court cases that deal with these individual components.
Especially with cases related to slavery you may find even more secondary literature in the library or on the internet.