  We prove that sufficiently low-entropy closed hypersurfaces can be perturbed
so that their mean curvature flow encounters only spherical and cylindrical
singularities. Our theorem applies to all closed surfaces in $\mathbb{R}^3$
with entropy $\leq 2$ and to all closed hypersurfaces in $\mathbb{R}^4$ with
entropy $\leq \lambda(\mathbb{S}^1 \times \mathbb{R}^2)$. When combined with
recent work of Daniels-Holgate, this strengthens Bernstein-Wang's low-entropy
Schoenflies-type theorem by relaxing the entropy bound to $\lambda(\mathbb{S}^1
\times \mathbb{R}^2)$. Our techniques, based on a novel density drop argument,
also lead to a new proof of generic regularity result for area-minimizing
hypersurfaces in eight dimensions (due to Hardt-Simon and Smale).
