Generally speaking, federal regulations are interpreted in the same ways that statutes are interpreted. The only real difference is that courts tend to give stronger effect to precedents interpreting statutes and federal regulations, which can be changed by elected officials, than to precedents interpreting the U.S. Constitution, which is extremely difficult to amend.
Statutory interpretation (including regulatory interpretation) is a whole sub-field of law and can't be easily summed up in a few sentences. There are probably a dozen or two maxims that are used to do so in particular circumstances, some of which are basically contradictory.
The big issue in federal regulatory law, however, and it is a huge one, is not actually about how federal regulations are interpreted. It is about how federal statutes are interpreted by federal regulations.
For forty years and six days, the Chevron doctrine (named after the U.S. Supreme Court case, Chevron U.S.A., Inc. v. NRDC, 467 U.S. 837 (June 25, 1984) that established it) held that federal courts should defer to federal agencies regarding how federal statutes are interpreted, especially when those agency interpretations of a statute are reflected in a duly adopted federal regulation.
This all changed when the U.S. Supreme Court overruled the Chevron doctrine in the case of Lopez Bright Enterprises v. Raimondo, No. 21-451 on June 28, 2024. See generally here.
Now, U.S. federal courts do not have to defer to a federal agency's interpretation of a statute that it is charged with enforcing, even if it is expressed in a duly adopted federal regulation.
Basically, federal regulations interpreting federal statutes are now only enforceable when a federal court, engaging in de novo review of the statute (i.e. review fresh with no deference to federal agency interpretations as expressed in the regulations), no matter how long it has been understood to be the controlling interpretation of that statute, mean exactly what the government say that it does. So, what a regulation says is now, to a significant extent, in a wide swath of cases, irrelevant.
Another recent case U.S. Supreme Court decision, Corner Post, Inc. v. Board of Governors, No. 22-1008 (July 1, 2024), it effectively eliminated the six year statute of limitations for facial challenges to federal regulations. This means, functionally, it is never too late to contest the process by which a federal regulation was adopted, and also overturns settled understanding of the meaning of the statute of limitations in question.
