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 at https://supremecourt.nebraska.gov/filing-petition-set-aside-criminal-conviction-county-court.
The value of expunging a criminal record has been called into question in recent years because of the advances in information technology which allow almost anyone access to all types of personal history. For a discussion of this topic go to: http://www.npr.org/templates/story/story.php?storyId=114276194.
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.