There are a lot of ways to study writing: the process (how is it done?), the product (what does it mean, what does it say, how does it say it?), the people involved (what do readers and writers do? How and why?).
There are also a lot of scholarly traditions which study writing: literary studies, Rhetoric and Composition, discourse analysis/discourse studies, applied linguistics, TESOL/ESL writing, etc.
I sometimes have trouble picking which one I'm coming from. For the most part I would say I'm a sociolinguist who studies writing, or a writing specialist who approaches the subject from a sociolinguistic perspective; that is, I am interested in how writing operates in society, or how social life and practices of writing (specifically the nitpicky ones -- grammar, lexis, etc) intersect.
I'm interested in how beliefs about grammar and sentence structure -- the way we conceive what is 'good writing' at the linguistic level -- affect the ways people perceive and judge writing and writers -- mostly in educational settings, but not always. (For example, I love pursuing nonstandard turns of phrase that show up in various places, like "every corn has two sides," which appears 79 times on the internet, figuring out how people are using them, and perhaps why, and how other react.)
That's pretty much what I do, scholarly-wise.