I don't think a contempt order for a civil child support case will show up in a criminal record search in North Carolina.
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It is my understanding that "civil" contempt does does not show up on a "criminal" background check; one found in civil contempt has committed a civil wrong and not a criminal violation.
However, a civil contempt may appear on a general background check.
You may want to consult a lawyer in your local area for further discussion.
This post should not be considered legal advice or advice that creates an attorney/client relationship. Readers should consult an attorney