There are two places to register a script. The US Copyright Office and the Writer’s Gild of America. Some people do one or the other and some people do both. Because you need to register each new draft in some cases, it can become somewhat expensive. However, it is good insurance.
As someone who has been sued for copyright infringement* I have a ton of opinions on this topic.
If you option your own material or the material from another writer, it is essential to keep the registration numbers files away somewhere safe. Side note – it is considered bad form, and a sigh of paranoia to put the number on a script.
If you use Final Draft to write your screenplay you can register with the WGA directly from through software. You just need wifi and a credit card. To do so, open the script in Final Draft, click on file. From the drop down go to the very last item which says, “Register”. From there follow directions. The cost is $25.
If you do not use Final Draft you can go to this link to register with the WGA: https://www.wgawregistry.org/registration.asp
Copyright registration currently costs $35-$55 for online applications and $85 for paper applications. If you are registering a copyright online, you may pay by credit or debit card or by an ACH transfer via Pay.gov. If you register by mail, you can pay by check or money order.
Using the U.S. Copyright Office’s online electronic registration system usually takes about three months. The average processing time for hard-copy registrations is 10 months.
I say be safe. Register early and often.
*(a nuisance suit filed to try to obtain the rights to my screenplay. Spoiler alert: I won! Read all about it: leagle.com/decision/infdco20131218b01