What are the other common approaches to study syntax?
Note: the source is an example question from the on-topic question list in Area51.
If you want to go in depth, check the links. To name a few:
and many many more.
I will add that in NLP/CL circles, Chomsky's approaches (Principles & Parameters, Minimalist Program) seem to be mentioned only for historical reasons. Chomsky's approaches have been superseded and seem to have been almost completely abandoned. Though dependencies are not exactly a grammar formalism, they seem to be far more commonly used than all of the above in NLP/CL circles. Having said that, I believe that many psycholinguists still use Chomsky's systems for their work.
You may want to check out Head-driven Phrase Structure Grammar, which is a non-derivational generative grammar, developed at Stanford beginning in 1987.
See: Pollard, Carl; Ivan A. Sag. (1994). Head-driven phrase structure grammar. Chicago: University of Chicago Press.
Don’t forget the “no particular approach” approach.
Haspelmath: “If there are no frameworks, then what should I teach my students in syntax classes? My answer is: The best syntax class is a field methods course, and the second best syntax class is a typology course.” - Framework-free grammatical theory.
Briefly, from Wikipedia:
In RRG, the description of a sentence in a particular language is formulated in terms of (a) its logical (semantic) structure and communicative functions, and (b) the grammatical procedures that are available in the language for the expression of these meanings.
Automodular Grammar (Sorry, no Wikipedia page)
A framework by Jerrold Sadock in which each module of language (Syntax, Semantics, Morphology, etc.) is completely independent of the others.
There are formalized dependency-based grammars, such as Meaning-Text Theory or Functional Generative Description. A simple Google search will give you links to papers and books.
AUG - Applicative universal grammar, by Shauman. It somewhat mix of chomsky ideas with semiotic approach.