What cases go directly to the supreme court rather than lower court?

Most of the US Supreme Court appellate cases originate in one the Circuit courts (U.S. Court of Appeals for the [Specific] Circuit), which is the appellate court between the District (trial) Courts and the US Supreme Court. These are not to be confused with the US Circuit Courts that existed between 1789 and 1911, which were primarily trial courts.

On rare occasions, cases may go directly from District (trial) Court to the Supreme Court on direct or expedited appeal, bypassing the Circuit Court. For example, United States v. Nixon, (1973), the dispute over whether the President was required to release the Watergate tapes to the Special Prosecutor, was appealed directly from the District Court. The flag desecration cases tried under the Flag Protection Act of 1989 were appealed directly to the US Supreme Court because Congress had specifically mandated that in the legislation.