Is it required or recommended that the property owner's spouse sign the mortgage, even though they are not an owner?

This is going to depend on your state law, and may depend on whether the property is the homestead or not. You should post in your state's forum. Some states require both spouses to sign, even though only one owns the homestead. Additionally, you will probably want any exemption from seizure waived (if permitted by state law) and often that requires both spouses to sign even though only one owns.

Should you just get the spouse to sign in every state?

Not unless they are required by law to secure the mortgage as it would be deemed a discriminatory practice based on marital status.