I applied to immigrate to America for my mother. Does she need a single certificate or a divorce certificate?

You are an American citizen, and your application is for your mother. Actually, it doesn't matter, as long as you can provide mother-child relationship.

For example, household registration book, DNA identification, because you are not applying for your wife.

Ask your mother to provide a divorce certificate at most, and then go to the civil affairs bureau where the household registration is located to open a single certificate and send it to the notary office for notarization. Do it.