In many situations, there is little benefit in a race to the courthouse. Sometimes it can make quite a difference what the status quo is when a divorce begins however. Who lives where, and where the children live, and the like can be important factors to freeze in time with court filings in some cases.
Another issue of importance can be which county a divorce is filed in.
A divorce (in Oregon) must be filed in a county where one spouse lives.
However, if the couple is split, and one person is living in Portland while the other is living in Benton County, for example, it can make quite a difference in convenience, and in cost, if the court that handles the divorce is in the county where you live, and where your lawyer practices, rather than in another county two hours away.
Some people think that whoever serves the other side with papers first is the one who gets to choose where the case will be heard.
This is incorrect.
The person who files their case in court first is the one who gets to be the Petitioner, and the court where they filed is the court that will hear and control the case. This is true even if the other side also files for divorce somewhere else, and even if the other side is the first one to achieve service of process.
Some years ago there could be a question when a case was actually filed, since many courts only date stamped a filing. When there was a race to the courthouse, in those days, the lawyer would get the court clerk to add an actual time as well. Now the time of filing, to the second, is automatically added for every case filed in court in Oregon, so these disputes are easily resolved.
If there have been two cases filed, the side which filed the second case generally is responsible for dismissing the second case, since there is already a case pending.