I have far too much material and tons of links on this question, so rather then attempting to be exhaustive I'll list a few of the more representative approaches to classification, and provide google-able phrases in bold. A more exhaustive collection of classifications can be found at:

http://www.ipl.org/div/farq/plot...

(note my list below is not a subset of the link above, but kind of a smaller, overlapping set)

1. "Stories that are classifiable are not real stories" as a normative stance. Some people call this "particularism." You can view this as either cardinality 0 or cardinality [math]\infty[/math]
2. Cardinality 1: The Freytag Triangle and Joseph Campbell's Monomyth are universalist models.
3. Cardinality 2 is basic Aristotle: tragedy and comedy
4. Christopher Booker has a 7-pattern classification: overcoming the monster, rags to riches, the Quest, comedy, tragedy, rebellion, mystery. Cardinality 7 is popular, there are many other 7-item lists.
5. Georges Polti has a classification of 36 plot types
6. It is worth examining the interesting life of Wycliffe Hill, dubbed the "King of all Formulas" by Slate: http://www.slate.com/id/2221392/ to understand how theories of narrative end up influencing narrative practice. Besides this, Campbell of course, has been influential.

Lately, there's been an attempt to think at the situation-atomic level rather than entire plots. Modular stories if you like. http://tvtropes.org, which has been mentioned, is a gold mine.

These things are valuable to ponder, but try not to take them too seriously.

I'll blog about this at length at some point, and a chapter of my book deals with narrative rationality (though classifying fiction is not the point in my treatment), but if you can't wait, and want a list of links to read, message me.

There are also non-Western narrative aesthetics/rationalities, but getting into those would make this way too long.