Copyright is a protection that is inherent for all creatives under US law. If you create something, your work is automatically copywritten. So what most people are talking about when they ask about should they copyright their music, they're talking about purchasing copyright protection, which is something that you can file and pay a fee for with the government.
Purchasing copyright protection involves filing an application and paying a fee to the appropriate government agency, typically the United States Copyright Office. While registering your copyright is not mandatory, it offers several advantages:
Legal Protection: Registered copyrights provide a legal basis for pursuing legal action against individuals or entities that infringe upon your intellectual property rights. This includes unauthorized use, reproduction, distribution, or performance of your music.
Presumption of Ownership: Copyright registration creates a public record of your ownership of the music, making it easier to prove your rights in case of a dispute. This can be particularly crucial when asserting your rights against larger corporations or entities.
Statutory Damages and Attorney's Fees: In the event of copyright infringement, registered copyright holders may be eligible for statutory damages and attorney's fees, providing stronger incentives for pursuing legal action.
Global Protection: While copyright protection is automatic in many countries, registering your copyright in the US can strengthen your legal standing internationally through various international treaties and agreements.
While these advantages sound great, as long as you can prove in court that you created the work (in the case of recorded music, providing original source files) you can win a case in the unlikely event your copyright is violated.
If you like to be very cautious, copyright your releases but I don’t think it’s the best use of an independent musician’s budget.