The secondary fan $Σ(k,n)$ is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices $Δ(k,n)$. We find new infinite families of rays of $Σ(k,n)$, and we compute the fans $Σ(2,7)$ and $Σ(3,6)$. In the special case $k=2$ the fan $Σ(2,n)$ is closely related to the metric fan $\mathop{MF}(n)$, which forms a natural parameter space for the metric spaces on $n$ points. So our results yield a classification of the finite metric spaces on seven points.