I´m a beginner on narrative studies. But among things I´ve read up to now, everything will depend on the way you define narrative. I will cite some view I´ve been dealing with.
There are the classical narrative studies. I could mention here Labov studies.
He focused on prototipical narratives and basically on the internal structure of a narrative.
There are the ones who work with small narratives, and take into consideration everything that was not considered in the classical studies. I could cite Bamberg, Georgakopoulou.
Similar to Georgakopoulou, there are people who focus on external aspects, such as social effects, of storytelling. We could think about people who work with sociolinguistics, here. There is a very good book - "The sociolinguistics of narrative" - available online. Here the analysis could be related to sociolinguistics theory.
I suggest Blommaert for critical sociolinguistic analysis.
There are other people who besides focusing on external aspects call attention to the cultural aspects. They understand narrative as performance. I could cite Richard Bauman and Charles Briggs. They are anthropological linguistics and explore the ways culture and performance are related to each other. This trend can also involve sociolinguistic theory for analysis.
A couple a days ago, I read some chapters by Gubrium e Holstein who discuss very deeply interview activity and define it as a narrative practice. I´m looking forward other sources of reading like this. They also propose a critical analysis concerning relations of power, cultural repertories, authenticity and ownership in narrative work. They have many books on the subject. Some chapters can be found on the web. And here again, the analysis can also be done in terms of critical sociolinguistics.
I´m not sure if I answered your question, but at least I hope I may have brought some perspectives for you. Best wishes.
What I have found most helpful is Naomi Quinn's approach: When she was faced with analyzing years and years of interview data about marriage, she focused on three things: metaphor, reasoning, and key words (Quinn 2005). I have also used this approach in my own field work with proverb enactments.
“How to Reconstruct Schemas People Share, From What They Say.” In Finding Culture in Talk: A Collection of Methods, edited by Naomi Quinn, 35-82. New York: Palgrave Macmillan.