The court offer opinions on about .8% of the cases and responds to about another .5%
The US Supreme Court received 7,738 petitions for Writ of Certiorari (requests for case review) in the 2008-2009 Term (the most recent year for which information is available). The Court estimated it had received more than 10,000 petitions last Term, but the number was revised downward in the Chief Justice's year-end report.
According to Court literature, the justices hear oral arguments for approximately 100 cases per Term, and issue full opinions for 75-85 of those. They write orders for another 50-60 cases reviewed without argument.