If a student loan is discharged, is the discharged amount taxable?

Not in all cases.  The general rule is that, if a debt you owe is discharged, the amount discharged now becomes income to you because you do not have to pay back the debt.  I.e., you received money and used it, but now you do not have to pay it back, which is the basic definition of income.  This is why when the bank forecloses on a home and writes off the remaining mortgage, the owner now has that much taxable income.  Several years ago, the IRS forgave student loans that were written off due to two college chains going bankrupt.  Revenue Procedure 2020-11 now extends that relief to a wider range of student loans.  If the federal student loans were discharged by the Department of Education under the Closed School or Defense To Repayment discharge process and in a few other cases, the recipients of the loans will not have to declare that write-off amount as income.