The United States Supreme Court has just created a new tool in beta for searching the text of oral arguments back to 1950. As of today, users can search the text of 1,431,579 individual statements in 6,709 oral arguments. Users can search by search term, search type (e.g., case name, case number, petitioner, respondent), Court Term (e.g., 2015), or speaker.
Audio recordings and transcripts of recent U.S. Supreme Court cases can be found on the bottom of the page here. Since 2010, the U.S. Supreme Court has archived all audio recordings of oral arguments online here. Same-day full transcripts of oral arguments since the year 2000 are also archived online at the U.S. Supreme Court's web site here. More Court information on oral arguments is available here.