As noted by the Colorado Supreme Court's Chief Justice Directive 05-01, any person can contact the local court to obtain case information, including case number, with a few exceptions. A criminal defendant can also contact her attorney or public defender for case information.
As established by Colorado, California and most states, any person can contact the court in which a case is being adjudicated to find out case information, including case numbers. The Supreme Court of Colorado's Judicial Department issued a formal policy stating that the public has the right to access court information, including case numbers, on all but a few cases. Some of the exceptions include juvenile cases, mental health cases and paternity cases.
Other state departments can offer searches when the exact court is unknown. For example, the Florida Department of Corrections has a free offender search page. The result shows information about the offender, including the case number of the criminal case. New York offers criminal records searches that allow searching for a person by exact name and birth date for a fee. This report includes case numbers, if any, though the website notes that any information on juvenile cases or cases that convert to juvenile cases does not appear on the report.