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$c=1$ rational CFTs have been completely classified, and the full list was first given in http://www.sciencedirect.com/science/article/pii/0550321388902490. There are infinite number of them. This should not be too surprising, since free compact bosons with different compactification radius already give you infinite examples. The rest are mostly orbifolds of ...
It can be rather involved. A lot of technical progress as been on this subject leading up to the modern conformal bootstrap work. Something you can exploit is that these functions should behave like correlation functions and thus are eigenfunctions of the conformal Casimir. That gives you differential equations which in some cases, especially in $D=2$ and ...