What is the fastest algorithm to establish whether a linear system in $\mathbb{R}$ has a solution?
Whether or not Coppersmith-Winograd is the "best" algorithm depends on your circumstances, of course. CW and algorithms like it are usually considered impractical due to high constant factors. Strassen's algorithm is more common in practice.

But since computational complexity is what you are interested in, CW was beaten quite recently.


Virginia Vassilevska Williams (2014), Multiplying matrices in $O(n^{2.373})$ time, Stanford University.


As far as we know, calculating the determinant of a matrix, or eigenvalue estimation, or anything else that could be used to determine if a matrix is singular or not are at least as complex as matrix multiplication.