The Volokh Conspiracy
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The Supreme Court publishes its opinions and orders in three primary locations.
First, the "Opinions of the Court" section lists all of the new "slip" opinions. (Do not confuse "slip" opinions with "bench" opinions). Generally, the Opinions of the Court included cases for which certiorari was granted, and were scheduled for oral argument. Even if a case is dismissed as improvidently granted, it will appear on the "Opinions of the Court" page. (For example, see Emulex Corp v. Varjabedian). The "Opinions of the Court" page also includes per curiam opinions from cases that were not argued. (For example, a summary reversal, or a reasoned opinion that denies a stay).
Second, the "Opinions Relating to Orders" section lists opinions for "summary dispositions." These cases often arise from an application for emergency injunctive relief. Usually, the document begins with an order. For example, "The motion to lift stay is denied." That order is followed by an opinion, perhaps an opinion respecting, or dissenting from, the order.
The Court offer this explanation:
Opinions may be written by Justices to comment on the summary disposition of cases by orders, e.g., if a Justice wants to dissent from the denial of certiorari or concur in that denial.
These opinions are generally formatted in the same fashion as slip opinions. My understanding is that the Reporter of Decisions reviews these opinions related to orders before they are published.
Third, the Court will sometimes publish opinions on the "Orders of the Court" page. Generally, this page will only unsigned, per curiam orders. For example, a petition for a writ of certiorari is granted or denied. Or an application for a stay is granted or denied. These orders are not published in the same format as opinions. My understanding is that the Reporter of Decisions does not review these orders before they are published. Though, they ultimately wind up in the U.S. Reports, and are ultimately vetted.
The miscellaneous orders will also note when there is a dissent from the order. For example, yesterday's order in RNC v. Common Cause of Rhode Island reflected that three Justices would have granted the application. And on occasion, a Justice will include a brief sentence explaining the grounds for dissent. For example:
18-6882 CRAIN, WILLIE S. V. FLORIDA The petition for a writ of certiorari is denied. Justice Sotomayor, dissenting from the denial of certiorari: I dissent for the reasons set out in Reynolds v. Florida, 586 U. S. ___ (2018) (Sotomayor, J., dissenting).
But if a Justice wishes to actually write a separate, signed opinion, the case will usually be filed under the second section, "Opinions relating to Orders." In other words, the dividing line between "Opinions relating to orders" and "Orders of the Court" is whether a Justice issues a signed opinion.
Still, there are outliers. In some 11th-hour capital cases, the Court will issue what is in effect an "opinion" in the "orders" section. That way, the opinion can be quickly released. The Reporter of Decisions can subsequently edit and format it. For example, in March 2017, Justice Breyer issued a three page dissenting opinion in Ruiz v. Texas. That opinion was initially included as part of the Court's order. Subsequently, that same dissent was published on the Opinions Relating to Orders Page, with the proper formatting.