Federal Rules of Civil Procedure 5.2(a) (Privacy Protections for Filings Made with the Court) states that any court filing "may include only" the last 4 digits of a person's social security number and just the year of a person's birth.
A federal district court in Minnesota (Engeseth v. County of Isanti) recently sanctioned an attorney $5,000 for including the full social security numbers of 179 people in an affidavit filed with the court and discussed the serious threat of identity theft.