  We study the small scale distribution of the $L^2$ mass of eigenfunctions of
the Laplacian on the flat torus $\mathbb T^d$. Given an orthonormal basis of
eigenfunctions, we show the existence of a density one subsequence whose $L^2$
mass equidistributes at small scales. In dimension two our result holds all the
way down to the Planck scale. For dimensions $d=3,4$ we can restrict to
individual eigenspaces and show small scale equidistribution in that context.
  We also study irregularities of quantum equidistribution: We construct
eigenfunctions whose $L^2$ mass does not equidistribute at all scales above the
Planck scale. Additionally, in dimension $d=4$ we show the existence of
eigenfunctions for which the proportion of $L^2$ mass in small balls blows up
at certain scales.
