Undecidable unary languages (also known as Tally languages)
Note that $\{1\}^*$ is isomorphic to $\Bbb N$. There are uncountably many subsets of both $\{1\}^*$ and $\Bbb N$.

Perhaps you are confused with the fact that there are only countably many finite subsets of $\{1\}^*$ (and $\Bbb N$).