It depends on how the property is titled. You should look at the deed to the house, not necessarily the mortgage. You will also need to figure out if the property is in the trust. If the deed shows the property in the trust, then the trust documents will control what happens to the house upon your death.
The mortgage is a 2 part issue: part 1 is the note - you agree to pay back the note (money loaned to you to buy the house) on the house and part 2 is the "mortgage" which allows the lender to foreclose if you default on the note. Both parts are just commonly reffered to as the mortgage.
The mortgage has no bearing on how the house will pass upon death. The key is how the house is "titled" during life, and that will be reflected on the deed; most likley either a warranty deed or a quit claim deed. You would have gotten the deed at the closing of the sale on the house and you probably have not looked at it since.
Quite likely you own the house as husband and wife, which is a form of joint ownership, usually with rights of survivorship (JTWROS), and further in Michigan likely as tenants by the entireties, which offers creditor protection that mere JTWROS doesn't. JTWROS means that as joint owners die, the survivors own the property by operation of law. If the house is titled to you as Trustees of your trust, then your trust will control. If not then the form of ownership during life will control the disposition of the house. Incorrect title to assets during life (title that does not match estate plan wishes) is probably the primary reason that many estate plans fail and why trusts often still require a probate to change title to assets post mortem, as assets pass contrary to the decedents wishes by operation of law.
Regardless, as a rule, when one inherits they take subject to a mortgage.
I agree with the other responses that the TITLE to the property is the key factor and not the mortgage. It is relatively unusual, however, to have one spouse on the mortgage and not the other; particularly if the person not on the mortgage IS on the title of the property. (Doing so could create problems for the lender as the spouse would have rights that they might not be able to foreclose on.)
I would absolutely check the title to determine how the property is owned. If it is in both names, then it would pass to the survivor upon death of the first spouse. If it is titled in the trust, that would also be fine, as there would be no need for probate upon the death of the first spouse. (If the title is in both names, then there probably should be a ladybird deed to the trust so that probate would not be necessary upon the death of the second spouse.)
If the house is titled in your name alone, then you need to execute a deed that would either add your wife to the title or have title pass to your trust, upon your death, or both. This should be done as soon as possible, in order to avoid the need for probate, in the event of your death.
If you were to pass away and your wife is on the title to the property, but not on the mortgage, she will still need make sure that the payments continue to be made or the lander may still foreclose on the property. They are generally content to allow the mortgage to continue in effect, as long as payments are being made. They would be within their rights, however, to force your wife to re-finance. It depends on the bank and the circumstances as to whether or not they would do this.
If you are actually married (you referred to her as your wife) then she has rights under law to your estate. She may also have ownership rights to the property, particularly if her name is on the title. Normally, if a lender grants a mortgage, the Bank has a title exam made to determine who the owners of the property are and requires all title owners to sign the mortgage even if the bank does not require all of them to sign the note. Sometimes a lender relies only on the credit of one borrower but if property is pledged under a mortgage, the lender always wants all parties who have an ownership interest in the property to agree to the mortgage which gives the lender the right to look to the property for payment if the debt (the note) is not paid. Even if she has rights to your estate, if the loan is not paid and the lender's lien under the mortgage has been agreed to by all parties having an ownership interest in the property at the time the mortgage was granted, the lender will be able to foreclose the lien created by the mortgage. Of course, if the lender is being paid the required mortgage payments, the lender may not have any reason to foreclose. The lender's right to foreclosure is to assure that the lender is paid. The lender does not want the property unless there is a default in the payment of their mortgage. In that case the lender goes through the process to foreclose the mortgage in order to be able to sell the property, get the loan repaid to the extent of the remaining balance and then lend out the money again. I think you need to find out if your wife's name is on the title and if it was on the title before the mortgage was recorded. You will want to consult with a lawyer to verify your position in respect to the mortgage.