It's not unusual for spouses to continue owning the family home together after a divorce, especially where kids are involved. For example, if one of you wants to buy the other out but can't afford to do it all at once, you might agree that payments can be made over time while both of you keep an interest in the house. Co-ownership also an option in a weak real estate market if you believe things are going to improve. Or you might delay the sale until a specified event, perhaps your youngest child's graduation from high school. (This is called a "deferred sale.")
There are pluses and minuses to co-ownership. If the custodial parent can't afford to buy the other one out, then the obvious advantage is that the kids get to stay in the house anyway, providing an important sense of security and continuity for them. It can make a buyout possible by spreading payments over time.
The disadvantages can be pretty significant, though. Because you are both responsible for paying the entire mortgage, a credit report for either of you will show the entire amount of your mortgage. Having such a large debt on your record, especially if you are not living in the house anymore, can make it difficult to get credit for other purposes. You also bear the risk that your spouse will make late mortgage payments that will hurt your credit rating.
There's also a fair amount of accounting involved. You must decide how you will share the mortgage and upkeep expenses and who can take the mortgage interest deduction. For example, even if you pay equal amounts toward the monthly mortgage, you can agree that one spouse who would benefit more from it gets to take the entire mortgage interest deduction, in exchange for increased support or some other equalizing payment.
It also means that you must continue to be involved with your spouse. Of course, if you are parents, that's already true, so this may not feel like a big additional burden. But if you anticipate that it will make the emotional disentanglement more difficult, think twice before you agree to this long-term commitment.
You run the additional risk that the spouse not living in the house might have a change of heart later and want (or need) to sell sooner than anticipated. Your settlement agreement should set a specific time that the house can be sold—if it does, the agreement will govern. But anyone who's really determined to get out of an agreement can make your life miserable in the bargain—for example, by claiming that the agreement was entered into under duress and forcing you into a court fight over that issue. So if you think the decision might not stick, don't make it in the first place.
If you own the house together for a significant period of time after your divorce becomes final, you also risk losing the important tax benefit of IRS Section 1041, which is the rule that says transfers between spouses as a result of a divorce are not taxable. Section 1041 applies as long as the transfer takes place within a year of the divorce becoming final, or as long as it's "related to the ending of your marriage," which means it's made under a written agreement or order and occurs within six years of the date your divorce becomes final (after six years, you lose the tax benefit no matter what). So make sure you don't just make a handshake agreement. Make the agreement to keep the house a part of your written settlement agreement, and get the court to approve it so that it becomes a court order.
Finally, consider two important risks. First, what would happen if one spouse died while you were still co-owners? Each of you has the right to leave your share at death. If you've agreed that one of you will stay in the house until the kids are a certain age, you could also agree that during that period you'll each leave your share of the house to the other, so that the resident spouse can continue to stay as you planned. This requires that you both make wills immediately.
The second risk is that one spouse will be sued by creditors or file for bankruptcy. In either of these cases, that spouse's share could be seized, possibly even resulting in a forced sale. There's really no way to protect against this, so if you believe it's a meaningful risk, don't go the co-ownership route.