I'll give you an explanation and provide a worksheet to make your calculation. The worksheet can be found at the following link: Insolvency Worksheet - Publication 4681--Canceled Debts, Foreclosures, Repossessions, and Abandonmen... (go to page 6 when the PDF file downloads).
The basic idea is to calculate how much you had in debt, and how much you had in assets right before the cancellation took place. You are insolvent to the extent that the amount of debt exceeded the amount of assets. You will see on the worksheet all sorts of possible debts: credit card, mortgage, medical bills, car notes, student loans, and several others. The amount you owe on these combined debts are liabilities. Then, you figure out your assets, which are your cash and other tangible possessions. If your assets are greater than your liabilities, you are not insolvent. Your canceled debt is considered income and taxed unless it was discharged (cancelled) for bankruptcy. If you are in debt, then you are insolvent (basically in the hole) to the amount that your debt is more. (If your debts are 10,000, and your assets are 5,000, you are insolvent by 5,000).
Next, you compare the amount of debt cancelled to the amount of insolvency. Taking the above example, if your cancelled debt was 7,000, your 5,000 insolvency would reduce the amount of debt treated as taxable income down to 2,000. 2,000 would be taxed as ordinary income in this situation. If your cancelled debt is 4,000, then your 5,000 insolvency means that none of the cancelled debt is taxable.
In cases where insolvency is used to reduced cancelled debt income, you will file with form 982. Please see this FAQ for additional information: https://ttlc.intuit.com/replies/3300645
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