Can a person in bancruptcy file a small claims action against someone even though it has not been approved by the trustee?

If the claim existed before the bankruptcy was filed then it needed to be listed in schedule B as a potential asset. Generally schedule B can be amended. Depending on the exemptions you have available in your state, you may be able to exempt the entire claim and keep the proceeds. If the amount of the claim exceeds the exemptions, the trustee may want to pursue the claim on behalf of your creditors but if it is a small claims matter, they may not want to bother. One other point. If you do not amend and it was a pre bankruptcy claim the creditor could argue your claim against them is barred because you had already signed a sworn statement in a court proceeding that you did not have a claim.