The dual graph $Γ(h)$ of a regular triangulation $Σ(h)$ carries a natural metric structure. The minimum spanning trees of $Γ(h)$ recently proved to be conclusive for detecting significant data signal in the context of population genetics. In this paper we prove that the parameter space of such minimum spanning trees is organized as a polyhedral fan, called the MST-fan of $Σ(h)$, which subdivides the secondary cone of $Σ(h)$ into parameter cones. We partially describe its local face structure and examine the connection to tropical geometry in virtue of matroids and Bergman fans.