Nebraska has a law, Nebraska Revised Statute §29-2264, which allows a person to petition the court for an order setting aside, or nullifying, the conviction. Forms and instructions for filing a petition to have your conviction set aside can be found in the self-help section of the Nebraska Judicial branch web site.
Nebraska Revised Statute §29–3523 is another Nebraska law which describes circumstances in which a criminal arrest (as opposed to a conviction) record can be corrected by petitioning the district court in the county where the person was arrested.