Is there a classification of or work towards a classification of countable ordered sets? Is there a classification of or work towards a classification of countable ordered sets?
 A: Yes. The reference to get started is 

MR0662564 (84m:06001) 
  Rosenstein, Joseph G.
  Linear orderings. 
  Pure and Applied Mathematics, 98. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. 

The key idea is Hausdorff's notion of scattered order. These are the ordered sets that contain no copies of the rationals. They admit a detailed classification, essentially by an analysis reminiscent of the study of Cantor-Bendixson derivatives (more precisely, we can build up all scattered ordered sets by iterating "sums" of simple orders, the last link below gives more details). 
The key result is Laver's theorem that Fraïssé’s conjecture holds: The class of countable linear ordering can be quasi-ordered by embeddablity. With this ordering, the class contains no infinite descending chain and no infinite antichain.
(So, this is not a classification in the model theoretic sense, but it is more detailed and far-reaching than simply discussing a basis, which of course consists simply of $\omega$ and $\omega^*$.)
If you want to see how Hausdorff's analysis can be used, Laver's proof is the result to study. For an extension, see:

Continuous Fraïssé Conjecture. Arnold Beckmann, Martin Goldstern, Norbert Preining. MR2470199 (2010a:03021) Order 25 (2008), no. 4, 281–298. 

Hausdorff's analysis can be continued into a study of uncountable ordered sets, but now things get much more involved, and ZFC does not settle their basic structure theory. See this answer to a related question on the sister site for more details.
