First, I will divide my answer/comment into divorce and annulment. Divorce is a legal procedure. Annulment is a religious event. If you are looking for an annulment, you will be searching for a church record.
For divorces, you are going to be looking for a legal record, usually filed with a court of government agency.
Next, where such legal records will be found in the government records will vary by country and division within country. Mostly, I don't know enough outside the U.S. to make useful comments.
Within the U.S., divorce has always been a court based process, so you will be looking for a court record. It is a matter of family law, so it is within state jurisdiction, not federal. So you will be looking for state or territorial court records. Most states and territories divided the state up into counties and divorces would have been handled in the local court.
At this stage, things begin to vary from state to state. Some states did not divide up their court systems by the kind of case, but most did and do. Divorce would be a civil matter, not a criminal one. Most states also divided up their civil courts into one type with juries that would handle contract and tort and other trial by jury matters and another type that generally handled more personal or family matters that usually did not have a jury including probating estates, handling guardianships, and other matters relating to marriage, family and personal life. Divorce would usually go in this non jury court. It would often be called chancery or probate court, but I know of at least one state that called it a court of common pleas and one that called it a district court. More recently states call them family courts but I don't recall any back as far as you asked about called family courts.
A major problem here is that the indexing of court records is very limited especially for the time period you are asking about and the computerization of what indexes exist is very limited. Mostly you have to find the local courthouse to search for the records.
So, you have to know where the divorce probably occurred down to the county level and maybe the city level. Marriage law is a matter of local jurisdiction and the place where a couple lived is usually the place where they would have to go to court to get a divorce. But often with divorce, one or both spouses would move and there would not be just one place where they lived. You may have to look in the county where they last lived together, the county where the husband lived at the time of the divorce and the county where the wife lived at the time of the divorce. So generally, you also have to know when the divorce occurred.
If you are lucky and have all this information, you go to the local courthouse in that place today and ask them where the records for that time period and that court are and if there are any indexes. There were not many divorces in the times you mentioned, so at best, you may find an index that lists the names of the plaintiff or complainant. So you would look up the name of both spouses in that index and hope you find a reference to the divorce case. If no index or if you don't find anything in the index, then ask if there is a docket book for the court of that period and page through it looking for the docket page (like a table of contents) for the divorce case you are looking for. Often there is either no docket book or it was the personal property of the judge and has not survived. So then you are left paging through all the documents of the court of that period looking for the divorce degree.
If you don't find it, then you consider whether there is some other place one of the spouses lived at the time of the divorce that might have the case.
This is a very difficult process and very hit or miss for the time period you are asking about. Mostly, you are likely to find a divorce record only when someone in the family kept it and passed a copy down.
One place where you may find evidence of a divorce in the time you are asking about is in the records of a later marriage. Usually, at least one party to a new marriage had to fill out an affidavit showing the couple was free to marry in order to get a marriage license. Either in the affidavit, the license, or sometimes a marriage register for the county, there may be a mention of a prior marriage that ended in death or divorce.
I hope this helps.