Are you aware of CCGbank (Hockenmaier, 2003)? This is the largest-scale corpus of English text annotated with CCG categories, consisting of ~1 million tokens of text, and is derived semi-automatically from the Penn Treebank.
This was used by Clark & Curran (2007) in a line of work to train wide-coverage dependency parsers for English, which made use of the fact that CCGbank covers a broad swathe of English syntax.
While the journal paper linked to above covers a number of the constructions handled, see Section 3 of Julia Hockenmaier's thesis for a complete list of the constructions that receive CCG analyses in the corpus.
It does cover rarer syntax which is nevertheless required for wide-coverage English text, such as conditional inversion ("Had it existed then, ...") which it analyses by assigning a category like ((S/S)/(S[pt]\NP))/NP to the token "had", accounting for the inversion.
Is there a resource I can consult that has a big list/summary of different parts of speech of English with their corresponding CCG categories?
Note that syntax doesn't map cleanly onto parts of speech, but you can easily gather a list of all categories that receive a certain part of speech by running some scripts on CCGbank directly.
