When applying for a K-1 fiancé visa, both the sponsor and the fiancé(e) seeking a green card need to prove they each have the “legal ability” to get married.
For the U.S. citizen sponsor, proving the legal ability to marry is straightforward. If they have been married previously and are now divorced or widowed, they attach divorce or death certificates to their visa petition for each previous marriage. The divorce decree or death certificates need to be official copies with the stamp of the granting court or government office. If the U.S. citizen has never been married, no paperwork is required.
The fiancé(e) needs to provide the same evidence regarding the divorce or death of a previous spouse. This area is a little trickier since every country handles these civil matters differently. U.S. officials will look closely at these documents to determine authenticity. U.S. Citizenship and Immigration Services (USCIS) refers to this as “proof of the legal termination of the previous marriage of the beneficiary.”
For more information on what to provide, as well as the exact documents (which vary from country to country), see U.S. Visa: Reciprocity and Civil Documents by Country on the State Department website.