When can a case go straight to the supreme court?

The only cases that currently go directly to the US Supreme Court under original (trial) jurisdiction are disputes between the states. The Court may elect to hear cases brought against ambassadors under original jurisdiction at their discretion, but typically allow the US District Courts to hold trials for those cases.

Explanation

The U.S. Supreme Court functions primarily as an appellate court; however, there are a limited number of cases for which they have original jurisdiction. Article III, Section 2 lists the following types of legal disputes under the Supreme Court's original jurisdiction:

1. all cases between two or more States;
2. all cases affecting Ambassadors, other public ministers and Consuls;

Disputes between the states are heard exclusively by the Supreme Court, but Congress amended federal law to allow US District Courts to hold concurrent original jurisdiction over cases involving ambassadors, etc.