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.