NB: Failing these sources, to get copies of cert petitions in cases where cert was denied, patrons may need to contact the Supreme Court Clerk's Office at (202) 479-3011 and have the docket number. It is likely that the patron will need to have someone physically go there and copy the document.
Two main sources:
SCOTUSblog: Merit, amicus, and petition granted briefs from 2007 to date; PDF
US Supreme Court > Docket Search: most filings submitted, including merit, amicus, and petition granted briefs, from Nov. 13, 2017 to date; PDF
Other sources:
Lexis+: Briefs, Pleadings, and Motions > Federal > U.S. Supreme Court Briefs
Landmark Briefs and Arguments of the Supreme Court of the United States: Constitutional Law: Online via HathiTrust and KF 101.9 L35
Office of the Solicitor General: All briefs from the OSG 1998 to date (PDF); selected older briefs 1982 to 1997 (html)
ProQuest Supreme Court Insight: Records and briefs from 1975. The collection also includes paid cases from 1975 to present in which the petition for writ of certiorari was denied. For the current term, the database will be adding material on an ongoing basis while a case is pending before the Court. Content associated with each case is compiled on a dynamic page organized to facilitate understanding of the judicial process and is also retrievable on a document by document basis.
Appellate case files of the Supreme Court of the United States, 1792-1831:
U.S. Supreme Court Records and Briefs Digital Archive (MOML):
Westlaw: Briefs > U.S. Supreme Court:
U.S. Supreme Court Records and Briefs (1960 to 2010):
U.S. Supreme Court Records and Briefs (1832 to date): Microfiche holdings
The joint appendix or appendices includes the relevant docket entries; any relevant pleadings, jury instructions, findings, conclusions, or opinions; the judgment, order, or decision under review; and any other parts of the record that the parties wish to bring to the Court's attention.
Lexis: joint appendices beginning in January 1979, with selected coverage from 1936.
Westlaw (SCT-JA): Selected joint appendices from 1982 to present.