A well-known result from descriptive set theory says that for any countable ordinal $\alpha$, the set of reals that code a well-ordering on $\omega$ with ordertype $\alpha$ is a Borel set. I wonder if such sets have positive measure or are non-meager.
An observation that might be useful: given a well-ordering $R\subseteq (\omega\times\omega)$ which has ordertype $\alpha$, any permutation of $\omega$ will generate an isomorphic relation. So the cardinality of the set of codes of $\alpha$ is $2^{\aleph_0}$. But I am not sure how to use this fact.