We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in the group $G_n = C_{2^n}\times C_{2^n}$. We use our algorithm to obtain all of these PDS in $G_n$ for $2\leq n\leq 9$, and we obtain partial results for $n=10$ and $n=11$. Most of these PDS are new. For $n\le 4$ we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in $G_n$ appear to grow super-exponentially in $n$. For $n=9$, a typical canonical coloring represents in excess of $10^{146}$ group-inequivalent PDS, and there are precisely $2^{520}$ reversible Hadamard difference sets.