(For our many international clients, please consult your domestic copyright office for details.)
In the United States, the U.S. Copyright office does have a field to supply your ISRC code(s) at the time of filing for your copyright. While not (yet?) mandatory, it certainly is helpful in making your registration as complete as possible. The field on their form is called “International Identification Number” You first pick ISRC from the “International Standard Number Type” drop-down and then enter your ISRC code for the recording that you are registering for copyright in the “International Identification Number” field.
We also recommend that when you upload your audio files to the copyright office, where possible, you upload the fully-embedded files. These are the versions that have the ISRC Code and all other META data already inside the audio files.