How does the Court system in Montana work for Divorce?

There are 56 District Courts in Montana.  These courts are split into 22 Judicial Districts.
District Courts are courts of general jurisdiction.  This means that these courts process all felony cases, all probate cases, most civil law cases at law and in equity, naturalization proceedings and various writs.  Most importantly, to you, is that District Courts process divorce, as divorce is civil law.  So when you file for dissolution (divorce), you file in District Court.

Flathead County is the Eleventh Judicial District Court, which currently has 4 District Court Judges.
If you lose your case, you may want to appeal it, in hopes of a better outcome.  An appeal from a District Court in Montana (including Flathead County), goes directly to the Supreme Court of Montana, the highest court in the state.  Montana is one of ten states in the U.S. that does not have an intermediate appellate court.  This is how the Supreme Court of Montana receives civil and criminal appeals directly from Montana District Courts.

The Montana Supreme Court currently has a Chief Justice and 6 Associate Justices.  Although only 4 Associate Justices are provided for in the Montana Constitution, the Montana Legislature is allowed to increase this number to 6.  Each Justice is elected for a term of 8 years.  All Supreme Court justices must be United States citizens, and have resided in Montana for at least 2 years prior.  They must also have been licensed to practice law in the state of Montana for at least 5 years prior to being elected.

The Montana Supreme Court also has original jurisdiction (the power to hear a case for the first time, rather than the power to review a lower court's decision) for cases not heard in District Court if there are no facts in dispute and the cases presents only legal or constitutional questions.

The Montana Constitution grants the Montana Supreme Court broad authority over the entire states courts to ensure smooth and effective operation.  The Montana Supreme Court also regulates admission of attorneys to the State Bar, as well as the conduct of attorneys and judges.