Your first question has an easy answer. The differential of $\det$ is
$$\sum_{i,j}\hat a_{ij}{\rm d}a_{ij},$$
where $\hat A$ is the cofactor matrix. Thus a singular point is such that $\hat A=0_n$, in other words, it has rank $\le n-2$.

Actually, ${\bf M}_n({\mathbb R})$ can be stratified by the sets $R_0,\ldots,R_n$ of matrices of rank $k=0,\ldots n$ respectively. Each $R_k$ is a submanifold of dimension $k(2n-k)$. $R_k$ is homogeneous, in the sense that ${\bf GL}_n({\mathbb R})\times{\bf GL}_n({\mathbb R})$ operates transitively on it by $(P,Q)\cdot A=PAQ^{-1}$. The relative boundary of $R_k$ is $R_0\cup\cdots\cup R_{k-1}$. In particular, there is no removable singularity.