For this scenario we will assume and clarify that your Ex spouse is solely designated as the cardholder or borrower on the line of credit; you are not designed as either.
The answer is this: It depends on what was charged to the credit card, and what the line of credit was used for.
Under the broader law of contract, the lender can usually only go after the specific borrower whose name appears on the credit/loan agreement. Using the credit card as an example: If the credit card is in your Ex’s name only, then technically the credit card company (e.g. Visa, Mastercard, American Express, etc.) usually cannot go after you to pay off his or her debt. You cannot be held liable under a cardholder agreement you did not even sign.
However, the matter gets complicated if you and your Ex decide to separate or divorce. Under the Ontario Family Law Act, you and both presumed to be equally responsible for family-related debts that are incurred during your marriage. These typically include clothing, food, family vacations, etc.
In the context of settling your divorce-related affairs and calculating your respective Net Family Property figures, a court may divide up this debt equally, or it may make an unequal division depending on the circumstances.
So family-related debts that were charged to a credit card in your Ex’s name, or funded through a line of credit in only his or her name, can still become your responsibility.