The Supreme Court website maintains opinions for cases beginning in 1991 in a PDF format on its web site.
Opinions are first issued as slip opinions. Then, "preliminary prints" of the United States Reports (the official case reporter of U.S. Supreme Court) are printed.
When the Supreme Court has made its final corrections, the United States Reports bound volumes are printed (usally about 3 years).
United States Reports
contains final opinions of the
United States Supreme Court.
How to Read a US Supreme Court Opinion by the American Bar Association
If the case you are looking for was argued before 1991, the unofficial sources below will be useful for locating opinions.
NexisUni
U.S. Supreme Court decisions from January 1790 to present.
Cornell's Legal Information Institute: Supreme Court Opinions
This site provides a comprehenisve list of opinions back to 1990. In addition it includes many pre-1990 decisions that are considered historically significant.
FindLaw: U.S. Supreme Court Opinions
Provides Supreme Court opinions from 1893 to the present.
Google Scholar
Select Case Law and follow the instructions.
Justia
Here you can find opinions from 1790 to the present. You can also look for recent opinions by year and U.S. Reports volume number.
Supreme Court Database
Includes information about each case decided by the Court from 1953 to 2008. It also contains neat analytical tools.