There is not enough information to fully answer your question. Generally speaking, as you said, there is nothing that you can do about secured debt. If the debt does not get paid, the creditor can foreclose or repossess the security and you are no further ahead. Unsecured debt is not so clear-cut.
In cases where there is an estate, the creditor can make a claim against the estate and if the amount is not paid, may be able to sue for a judgment. Once the creditor has a judgment, they can collect against the real estate or other estate assets.
If there is no estate, the situation is much less certain for the creditor. I believe that they would still need to open an estate in order to have someone to sue. This involves some costs on their part, which they probably would not undertake, unless they are certain there is property to collect on, and if the debt is high enough to warrant the risk and the delay. In many such cases, the creditor will simply write off the debt.
Another possibility is that you might try to negotiate with them to wipe out the debt for a fraction of what is owed. I have had heirs negotiate the debt down to 10¢ on the dollar, or lower. It depends on the creditor and the amount owed. You might want to consider trying this before deciding what to do, however. If you can completely wipe out the debt and move on, it might be worth it, particularly if they are willing to cut you a good deal.
Best of luck to you!
You and your siblings will have to pay the mortgage if you want to keep the house or sell the house. Depending upon whether there is anything in your mother's sole name to be probated, you may not have to open and estate or pay the credit card. Even if you have to open a probate estate, there may not be enough in the estate after the priorities are paid, such as funeral expenses. Please contact me to discuss your alternatives in more detail.
All debts are enforcable under the written contract or agreement signed by the person who owes the money. If mom was the only signer on the card, then she was the only one liable for it. If there was a probate estate opened, then the card debt could be a claim against the probate estate. If no assets or property went to probate, then there is nowhere for that creditor (bank, card) to make a claim. They can't make a claim against assets you received outside of probate, by operation of law or beneficiary designation in life insurance or retirement benefit, etc. Assets in a grantor trust may be liable for her debts however.
If you don't have title to the house, and there is equity in the house, it will have to be administered in probate. This will expose the equity to the claim for the card.
You haven't given enough information for a definitive answer, but the alternatives discussed above may cover your situation.