This is another question that pops up every now and then in both online and real world conversations, so I thought it was a good subject for a blog rant!
As a disclaimer, this post is not legal advice. If you are in a situation where you find yourself in need of some, go to an attorney and make your own research - also because things can and will change from the date of this post. I will try and keep it up to date, but this post is simply something that I can point at when the issue of copyright surfaces one more time in discussions.
Also, this post is from the perspective of an individual singer/songwriter, band or small entity on a budget, not an a large organization consistently making large profits from music (in which case the answer will generally "yes", at least in the US).
First of all, let me start at the beginning: if you are making that question, you most likely have no idea of what you're talking about.
And since we are at it:
no, you cannot mail yourself a copy of your work as a copyright proof.
in most of the world you cannot "officially" register your copyright at all
even where you can, sometimes you don't have to do it and sometimes (like in the US) you don't have to do it beforehand, in order to be able to defend your right.
Ok, ok, I don't want to be mean. The question is legitimate, but it comes from a fundamental misunderstanding of what copyright is and how it works.
Let's have a look.
The first thing to understand is that copyright simply exists.
You don't have to do anything to create it.
It's that simple. In general terms, the moment you create an original work, you have copyright on it. That is to say, the right to control how your creation is used and (if applicable) copied, either directly or in other form.
Even if in theory it is a right that applies to any situation, in practice the right covers most often commercial uses - meaning that it's designed to ensure that, if money is made from your creation, you can get a slice of it (so that, ideally, you have an incentive to create more).
So what you are really thinking about is a situation when someone takes your creation and makes a ton money out of it... but passing it as his own and conveniently forgetting to recognize that you are the author.
That means that the notion of copyright becomes concretely meaningful only if:
Someone is making money out of your music (or you are making money out of theirs)
You (or someone else) concretely decide to contest that use, filing a suit - making a copyright claim
So the question of "should I copyright my music?" is really about "in case my song is picked up by someone who makes a ton of dough out of it, what can I do to ensure that I have a way to show it was mine and get some?"
A bit like a Schrodinger cat, the issue of copyright exists and does not at the same time, until someone decides to actively spend time, energy and money to go checking - i.e. enter a civil litigation.
And that's a can of worms.
First of all, while the idea of copyright is, give or take, the same everywhere, the actual legal implementation is absolutely not. Not even close. Every single country has its own legal framework (they may be similar, they may be not) and of course the degree of enforcement varies.
Most countries don't have a specific "registry of creations" for copyright enforcement purposes.
A copyright litigation in these places requires that you prove your claim, and any kind of evidence can be brought and is evaluated by whatever legal process civil claims are subjected to.
In each country there may be a legal tradition of accepting certain "proofs" more easily than others (judges like to look at precedents), but there's never a predetermined way that can ensure you that your claim is valid. You have - literally - to make your case.
In a few places, most notably the US, there is indeed such a registration office (the "Copyright office"), and you can apply to have your song registered there for some money.
Since 2019, the US Supreme Court has decided that having registered a work with the Copyright Office is a pre-requisite to start a litigation. But you don't need to have it done before you discover the violation - you simply need it done before you actually try to litigate it. It can take some time tough, so you may need to wait some if you do it afterwards. The Supreme Court decision also introduced the option to pay a fee to the CO to rush the process.
However, a registration in the US Copyright office is simply a pre-evaluated way of showing your claim and indeed provides certain advantages, when a litigation happens in US Courts.
If your litigation happens elsewhere, the US Copyright registration, while still being a strong proof of claim will simply be one of the proof types you can try to use. There can be copyright relationships, treaties and agreement between the US and other countries but you really need to check about the specific country.
Even before 2019, if you wanted to litigate a copyright claim in the US, the benefits of a filing with the US Copyright Office were:
you became eligible for "statutory damages", which means that, if you win the claim you don't have to prove the precise amount of financial damage.
as a consequence, it was easier to find attorneys willing to work for a percentage of the winnings at the end, instead of being paid for their work as the litigation goes. That's because if the attorney thinks he's gonna win, it's easy for him to predict what the amount will be and therefore his own profit.
already before 2019, certain court circuits in the US demanded a registration in order to litigate a copyright claim, others didn't. The US supreme court decision simply states that it is always necessary.
In other countries, there may or may not be any prerequisites to file a claim, so really you need to look at where the violation occurs and in which legal system you're gonna litigate it.
So if your country does not have a copyright registration office, what you do? How can you increase your odds in case someone in the future uses your music to make millions and refuses to acknowledge you?
Well, there's no bomb-proof answer, but first of all: no, you don't mail a cd to yourself!
That doesn't work at all, because it's far too easy to tamper with, and no sane court or judge would ever think of admitting is as evidence (and actually never has).
Your best bet is to actually publish your music, with an independent actor large enough that it has a chance to still be around when, in the future, the millions will be made. The publication date has a certain strength to establish the primacy of your claim.
It also helps to upload to YouTube or other services (the larger the better, because once again you'd like them to be still around at a possible future time of litigation) because they are independent from you: once uploaded, you cannot easily change the date of upload, and neither can your counterpart.
In most cases, if you are on a budget, that's your best bet. In the US, if you discover a violation, you can file a registration and - if possible at the time - maybe pay a fee to expedite things up so to be able to start a litigation for your damages.
It all boils down, however, to the fact that thinking about copyright is meaningful only if someone decides to litigate it.
And in all countries litigation is expensive and may take a long time. Most if not all of the landmark cases in music-related copyright ligation have taken years to be decided. That means that significant amounts of money must be involved to make it worth to raise a claim.
Unless of course you're filthy rich and can afford to do it at a loss.
So the answer to the question "Should I copyright my music" is it depends:
are you in the US? Consider the cost of filing a registration vs. the likelihood that someone actually is going to pass your music as theirs and make a ton of money with it. Even then, consider that you can register when you actually decided to litigate.
did you discover is there an actual violation of your copyright?
is there a significant amount of dosh made by violating it? Is it worth litigating?
is there a single entity that can be sued, that is making that dosh and not giving you your part?
My $10 is that, unless you are already a very established and profitable act, most often, simply publishing your music with a publisher or an aggregator (and use the synchronization and licensing agreements which come with it) is more than enough.
Should you to discover a violation by an US actor, and decide it's worth pursuing a litigation, you may always register there and then.
In the end, if you are in the US or elsewhere, it's up to you to balance the potential pros and cons.
But whatever you do, don't mail yourself a CD!