Finding an Uncomputable Function
Your definition is a complicated way to say that $S$ is the set of partial functions $X\to Y$ (the sets $A$ you choose are the domains of the partial functions).

Now, if you want to talk about computability, you have to be more precise on $X$ and $Y$: every element of $X$ and $Y$ have to be finitely described, so that they can be manipulated by algorithms.

We can for instance consider that $X=Y=\mathbb{N}$. Since they are countable and infinite, we just have to apply some bijections to get this.

Now, the framework you define is exactly the one of computability theory, you can define turing machines etc, and find out what the uncomputable are. Go here for more details.

For instance the halting function (which is in fact a function $\mathbb N\to \{0,1\}$) is uncomputable.
As for an example of subset of $S$ which contains only uncomputable functions, it is true as soon as the domain of the function is not a recursively enumerable set.
For instance there is no recursive function $NH\to \mathbb N$, where $NH$ is the set of indices of Turing machines that do not halt on input $0$.