  The global structure of the minimal spanning tree (MST) is expected to be
universal for a large class of underlying random discrete structures. However,
very little is known about the intrinsic geometry of MSTs of most standard
models, and so far the scaling limit of the MST viewed as a metric measure
space has only been identified in the case of the complete graph [5]. In this
work, we show that the MST constructed by assigning i.i.d. continuous
edge-weights to either the random (simple) $3$-regular graph or the $3$-regular
configuration model on $n$ vertices, endowed with the tree distance scaled by
$n^{-1/3}$ and the uniform probability measure on the vertices, converges in
distribution with respect to Gromov-Hausdorff-Prokhorov topology to a random
compact metric measure space. Further, this limiting space has the same law as
the scaling limit of the MST of the complete graph identified in [5] up to a
scaling factor of $6^{1/3}$. Our proof relies on a novel argument that proceeds
via a comparison between a $3$-regular configuration model and the largest
component in the critical Erd\H{o}s-R\'enyi random graph. The techniques of
this paper can be used to establish the scaling limit of the MST in the setting
of general random graphs with given degree sequences provided two additional
technical conditions are verified.
