Classifying infinite languages of encodings of halting Turing machines
There are certainly examples for all "levels":


It is easy to give an infinite set of encodings that is decidable.


   Consider all programs of the form f(x) = x + i -- definitely an infinite and decidable set.

Classic examples (think Halting problem) are semi-decidable but not decidable.


   $L = \{ \langle M \rangle \mid f_M(\langle M \rangle) = 1\}$

Other classic examples (think complement of Halting problem) are not semi-decidable.


    $L = \{ \langle M \rangle \mid f_M(\langle M \rangle)\uparrow \}$