according to Wikipedia conditioning is the rate at which a function changes in response to small changes in its inputs, and the condition number associated with the linear equation $Ax = b$ gives a bound on how inaccurate the solution $x$ will be after approximation. In the case of the function $$f(x) =A^{-1}x $$ . When $A \in R^{n\times n}$ has eigenvalue decomposition, the condition number is the ratio of the magnitudes of the largest and smallest eigenvalues.
Is there any similar concept for singular matrices? That is whether the ratio of the magnitudes of the largest and smallest singular values can reveal similar information?
As has been pointed out; the condition of a matrix is the ratio of the extreme singular values. Singular matrices will always have an singular value of 0, making its condition tend to infinity. On the other hand, the condition of a matrix builds an intuition for what the worst case relative error might be (again, infinite).
It appears to me that what you are after is some kind of average case relative error, which would require knowledge of the distribution of the solutions to your system.