Any finite linear ordering can be embedded in Q. Any countable model of Th(Q) is isomorphic to Q. Hence any countable linear ordering can be embedded into Q, by compactness and lowenheim skolem using the above, and the diagram of the l.o. in question.
So all countable well orderings (i.e. < omega_1). Of the top of my head (on my first coffee of the day), I'd say you'd have problems with point whose cofinality was greater than omega, since R is forst countable.