  We study the cubic defocusing nonlinear Schr\"odinger equation on
$\mathbb{R}^4$ with supercritical initial data. For randomized initial data in
$H^s(\mathbb{R}^4)$, we prove almost sure local wellposedness for $\frac{1}{7}
< s < 1$ and almost sure scattering for $\frac{5}{7} < s < 1$. The
randomization is based on a unit-scale decomposition in frequency space, a
decomposition in the angular variable, and - for the almost sure scattering
result - an additional unit-scale decomposition in physical space. We employ
new probabilistic estimates for the linear Schr\"odinger flow with randomized
data, where we effectively combine the advantages of the different
decompositions.
