I quickly scanned the answers and didn't find the following:

Square matrices of order n can be considered to be embedded in R^(nxn). The determinant is a continuous function of the entries of the matrix, so the singular matrices are det^(-1)(0) and are therefore a closed set in R^(nxn). Thus, there will be non-singular matrices arbitrarily close to a singular matrix in any convenient metric on R^(nxn). Of course, they will have pretty ugly condition numbers.

I quickly scanned the answers and didn't find the following:

Square matrices of order $n$ can be considered to be embedded in $\Bbb R^{n \times n}$. The determinant is a continuous function of the entries of the matrix, so the singular matrices are $\det^{-1}(0)$ and are therefore a closed set in $\Bbb R^{n \times n}$. Thus, there will be non-singular matrices arbitrarily close to a singular matrix in any convenient metric on $\Bbb R^{n \times n}$. Of course, they will have pretty ugly condition numbers.