  In this paper we study a neighborhood of generic singularities formed by mean
curvature flow (MCF). We limit our consideration to the singularities modelled
on $\mathbb{S}^3\times\mathbb{R}$ because, compared to the cases
$\mathbb{S}^k\times \mathbb{R}^{l}$ with $l\geq 2$, the present case has the
fewest possibilities to be considered. For various possibilities, we provide a
detailed description for a small, but fixed, neighborhood of singularity, and
prove that a small neighborhood of the singularity is mean convex, and the
singularity is isolated. For the remaining possibilities, we conjecture that an
entire neighborhood of the singularity becomes singular at the time of blowup,
and present evidences to support this conjecture. A key technique is that, when
looking for a dominating direction for the rescaled MCF, we need a normal form
transformation, as a result, the rescaled MCF is parametrized over some chosen
curved cylinder, instead over a standard straight one.
  This is a long paper. The introduction is carefully written to present the
key steps and ideas.
