So you made the right choice.

Regardless of whether your work is commercial or royalty-free, copyright will provide a smart legal backbone to you work. If you haven't read up yet, go check out the Copyright Basics section for a nice copyright primer.

Step 1: What copyright form should I get?
There are several different kinds of copyright that are offered by the United States Copyright office. Based on how you use the file, the type of copyright to file for will shift slightly.  Luckily, Flash will only get covered by two different kinds of copyright, depending on whether your Flash has interactive features or not.

My Flash Project has Interactivity or is a Game
If your Flash file is a game in which the code is more important to protect, then your best bet is to seek a Literary Copyright.  Because you are physically writing a body of text and compiling it into another text, your copyright is treated as a computer program, even though the Flash Player runs the file.  Games and interactive projects are protected as literary works.

If you are the only author, you can use the Short Form TX. If you are one of multiple authors, you should use the Form TX. The only difference between the forms is the amount you have to fill out.

My Flash Project is a Movie, Animation, or Moving Picture
If your work is an animation, film, stop-frame, or any other moving picture, you should seek a Performing Arts copyright.  Although your animation is in theory just a bunch of code, you are performing your Flash document on the web.  Thus, a Performing Arts copyright is more important. Even if your animation has a preloader (which is considered interactive content), you should log it as an Performing Arts copyright.

If you are the only author AND you were not sponsored or commissioned to make the work, you can use the Short Form PA. Otherwise, use the Form PA. The difference between the forms is the amount you have to fill out.

If there are other forms you need or think you might need because you have special circumstances, you can visit the Forms page for the United States Copyright office.

Step 2: Filling out the Forms
The forms come with detailed instructions printed above or around each question.

These forms come in interactive PDF format, meaning that you can type straight into the form without printing it out first. For the sake of clarity and concise writing, it is recommended that you type in the forms instead of writing in pen. If you do choose to write in the answers, make sure you use a black ballpoint pen.

After you are done filling out the forms, print them out. It is recommended that you print out the forms back-to-back instead of one-sided, to save paper and resources.

Step 3: Put together the package
In order for the copyright to be secured, your project must be deposited into the Library of Congress. Depending on whether it's an interactive project or animation/moving picture, you will need to enclose different things (make copies and remember to save a copy for yourself):

If your project is a game or interactive feature, you need to include:
A CD or DVD (protected in CD Case) with the final .swf files and final .fla files. Include an html file with the project on it for viewing purposes.
2 seperate printouts, stapled, of the first 25 and last 25 pages of your main body of code. If your code is less than 50 pages, then just send the whole thing.
The Short Form TX or the Form TX (see above for more info)
A check for $45.00, payable to "Register of Copyrights". DO NOT send cash!

If your project is an animation or film, you need to include:
A CD or DVD (protected in CD Case) with the final .swf files and final .fla files. Include an html file with the project on it for viewing purposes, as the copyright office needs a medium by which to view the Flash project.
A summary, press release, or some other document explaining what the animation or film is, printed out on paper.
The Short Form PA or the Form PA (see above for more info)
A check for $45.00, payable to "Register of Copyrights". DO NOT send cash!

Step 4: Send it Off!
Enclose all the goods in a box, not an envelope. All mail going to Capitol Hill is now being screened, and will be opened before reaching the copyright office. To better protect your paper items, use a small box.

It is recommended that you visit your local post office to send the package. This way, you can get the exact weight and cost for shipping, instead of guessing with stamps.

Mail to:

Library of Congress
Copyright Office
101 Independence Avenue, S.E.
Washington, D.C. 20559-6000

Remember to include your return address in the top left corner! If anything goes wrong, it would be terrible not to have it shipped back to you.

Step 5: Wait...

It should take on average 4 months to process your copyright. If there is anything wrong, they will email or call you and resolve the issue quickly.

Congrats! In the mail, you'll recieve your official certificate of copyright, perfect for framing!