Divorce doesn't make debts go away. The obligations live on after a couple parts ways, and both spouses are usually still responsible for paying them. Divorce decrees can address them, but decrees can't always force payment. Contracts and divorce fall into two separate areas of the law, and the areas don't overlap.
Your joint marital debts are contracts between you, your spouse and your creditors. A divorce decree can't nullify contracts, and your creditors don't care if you're no longer married. Two people signed for the accounts, and creditors are entitled to look to both for payment. This is true even if a decree states that only one spouse is responsible for paying them off post-divorce.
Many state courts assign premarital debt to the spouse who contracted for it before the marriage. This is "separate" debt -- it belongs to just one spouse. Separate debt does not refer to an account or contract signed for by just one spouse during the marriage. Divorce courts usually treat these accounts as joint marital responsibilities and allocate them between spouses for repayment. However, creditors usually have no legal right to go after the spouse who didn't sign for these loans. They can only pursue the spouse whose name is on the contract. Exceptions to this rule exist in the nine community property states.
Terms of Decree
Your divorce decree is capable of providing some damage control against joint debts, if you take care to word it correctly. If your decree assigns a certain debt to your spouse and she doesn't pay it, your credit is blemished as well. The creditor can sue you for the money if your spouse doesn't pay. But you can include provisions in your decree for this eventuality, such as that if your spouse defaults on an assigned debt, and if you must pay it yourself to protect your credit, she must reimburse you. Although your decree has no power over your creditors, it is binding on you and your spouse, so family court can enforce it and order your spouse to repay you.
It's usually easiest not to allow joint accounts to remain open after your divorce is final. Some couples pay them off and eliminate them at the time of the divorce. If sufficient cash isn't on hand to achieve this, you may be able to sell some assets. Another method is to have your spouse transfer the assigned debt into another account in her sole name. For example, if she has a credit card with a sufficient available balance, she can use that card to pay off a joint one. Then only she is contractually liable for the debt. Bankruptcy usually isn't a solution, unless you and your spouse file together. It doesn't solve the problem of jointly signed-for accounts, because after one spouse files for bankruptcy protection, the creditor will just pursue the other spouse for payment.