There is no court-derived rule on selection of cases for publication at the trial level. Chambers staff can, however, submit judicial opinions directly to vendors to request inclusion for publication.
Note that there are specialized case reporters, apart from the Federal Supplement, that publish trial-level cases. These include the Bankruptcy Reporter, Federal Rules Decisions, and etc. Typically, when one vendor selects a judicial opinion for publication other vendors will likewise re-publish. Also, if a case is selected for publication in the Federal Supplement it will not be additionally published in a specialized reporter (and vice versa).
Circuit Rule 36-2 governs the criteria for selecting appellate cases for publication. A written, reasoned disposition of a case not intended for publication is a memorandum. A written disposition of a case designated for publication is an opinion of the court. The criteria for publication includes: establishes, modifies, or clarifies rule of federal law; calls attention to rule that has been overlooked; criticizes existing law; has unique public interest; follows a remand or reversal by SCOTUS; or is accompanied by a separate concurring/dissenting expression where the author has requested publication. Dispositions designated as opinions are sent to the clerk’s office.