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In this paper we study a neighborhood of generic singularities formed by meancurvature flow (MCF). We limit our consideration to the singularities modelledon $\mathbb{S}^3\times\mathbb{R}$ because, compared to the cases$\mathbb{S}^k\times \mathbb{R}^{l}$ with $l\geq 2$, the present case has thefewest possibilities to be considered. For various possibilities, we provide adetailed description for a small, but fixed, neighborhood of singularity, andprove that a small neighborhood of the singularity is mean convex, and thesingularity is isolated. For the remaining possibilities, we conjecture that anentire neighborhood of the singularity becomes singular at the time of blowup,and present evidences to support this conjecture. A key technique is that, whenlooking for a dominating direction for the rescaled MCF, we need a normal formtransformation, as a result, the rescaled MCF is parametrized over some chosencurved cylinder, instead over a standard straight one.  This is a long paper. The introduction is carefully written to present thekey steps and ideas.

On the non-degenerate and degenerate generic singularities formed by mean curvature flow