I'm having problems coming up with a good beginning. I'm going to have to think it out for a while.
Note that beginnings are often a combination of things.
Opening Stasis: Shows things as they are before something significant changes. (As common as boy-meets-girl. LOTR. The Hobbit. Star Wars. If the characters are happy with their setting, the goal of the story is to return to the opening stasis (which may or may not happen). If the characters are not happy with their setting, the goal of the story is to escape the opening stasis and achieve transformation.)
In media res: Shows things in the middle of the action and events.
--Orientation: Shows action that introduces the characters and action. (James Bond movies. The goals are similar to that of the opening stasis beginning.)
--Disorientation: Shows action that introduces the mood. The characters and action are not explained. (Serenity. The goal of the story is to come to grips with reality. Either the reality is as complex for the characters as it is to the audience, or something about the assumed reality is not what it seems. The characters either accept or reject reality.)
Frame Story: Surrounds the main story with a second story.
--Inside: The main story and the frame story are in the same reality. (Harry Potter series. The main story undercuts the assumptions of the frame story.)
--Outside: The main story and the frame story are in two different realities. Story within a story. (The Princess Bride. The main story reinforces or enriches the assumptions of the frame story.)
Simple Story: The story begins with a simple plot which is undercut, reversed, or complicated. (The Dark Tower series. The Sixth Sense. Stories with a twist. The simple plot may or may not have anything to do with the main plot.)
Story-Pattern: The pattern of the introduction shows the pattern of the story as a whole.
--Self-similar: The beginning of the story takes a pattern repeated by the story as a whole. (Scooby-Doo, TV and movies. Spirited Away. Two approaches. In one, the audience loves the familiarity of the story. Often used in TV shows. The story before the first commerical break is something the main characters do all the time, with variations. In the other, a lesson is presented to the characters, which they do not learn. The rest of the story is the repetition of the lesson, with higher stakes.)
--Cyclic: The beginning of the story ties to the end. (The Lion King. American Beauty. The beginning reveals the end of the story, either directly or indirectly. The story often feels mythic.)
Backstory: The story is introduced by a past event. (X-men. An easy way to focus the story on the action of the plot rather than the idea of discovery of the truth. Can be a self-similar story.)
Squishy Intro: The story is introduced by an event that happens to a minor character. (Horror movies. Works in the same type of way as the backstory, but focuses on events where the backstory focuses more on character.)
Flash Forward: The story is introduced by an event that happens in the middle or end of the story but is not the ending of the story itself. (Fight Club. Swordfish. Clever action movies. Challenges the audience to try to guess the plot. Differs from a cyclic plot in that the events do not feel inevitable, and unpredictable things often happen after the story catches up to the first scene again.)
Things Which Will Be Revealed to the POV Characters Later: The story is introduced by an event that will not be revealed to the main characters until later. (Murder mysteries. Sometimes unusual events are disguised with normalcy. Easy way to introduce "mystery" to the audience. Often combined with a lie.)
Lie: The events in the beginning of the story are a lie or red herring.
Metafiction: The story is introduced by commentary on the story itself. (The Princess Bride. A Series of Unfortunate Events. A story about stories. Metafiction is like a frame story, but one in which the audience is to consider the events of the story other than they otherwise would.)
Cosmological: The story begins with an introduction to the universe itself. Discworld. Dune. Either I barely notice this kind of beginning or I hate it.)