We now have online access to the Gale database The Making Of Modern Law: U.S. Supreme Court Records and Briefs, 1832-1978 (description). This database contains nearly 11 million pages of official court filings - records and briefs from over 150,000 cases brought before the U.S. Supreme Court - offering the most complete digital collection of Supreme Court cases for this time period. It concentrates on the history of the Supreme Court from the final years of the court’s fourth chief justice, John Marshall, through the first 10 years of the court’s 15th justice, Warren Earl Burger.
In this database we have access to the full text of more than 350,000 documents, including appellant's and appellee's briefs, oral transcripts, and petitions for writ of certiorari. You can search for specific terms, or you can browse the collection by Author or case name : here's a screenshot of the documents available for Plessy v. Ferguson, for example. Please note, however, that this collection does not include the Court’s rulings, opinions or decisions.
The database is available at any Pitt computer.
Update: Please note that the database is available on the Barco Law Library home page under "Licensed Resources".