Have non-regular language classes of infinite words been studied?
We have the following MSO characterization of regular nested word languages:


  Regular languages over nested words are exactly the set of languages described by Monadic second-order logic with two unary predicates call and return, linear successor and the matching relation ↝.


The recommended reference on Nested Words and Visibly Pushdown Languages treats infinite words in section 8. Nested $\omega$-Words, and shows that the MSO characterization carries over:


  Theorem  8.4 (MSO characterization of $\omega$-languages).
  A language $L$ of nested $\omega$-words over $\Sigma$ is regular iff there is an MSO sentence $\varphi$ over $\Sigma$ that defines $L$.


Because often deterministic CFGs are actually visibly pushdown languages, this covers a huge portion of the practically relevant CFGs. Moreover, this even provides a context where considering $\omega$-languages is practically relevant, as explained in the same section of the reference.