Is there a noun for "unpaid debts"?
Bad debt
Debt from a credit sale that the creditor is unable to collect. Debt becomes bad debt when the creditor has made all reasonable efforts to collect the debt but has been unable to do so. …
(TFD Financial)
"I always entrust to a lawyer the case of a bad debt."
See also Doubtful debt (BusinessDictionary) and "The difference between bad debt and doubtful debt" (AccountingTools)
Good luck.