The condition that ann-by-nset of simultaneous linear equations over the reals have a unique solution is just that the determinant of the matrix of coefficients be zero. Since determinant is a polynomial in the entries of the matrix, the locus of determinant equal to zero is a set of measure zero.

The situation changes over a finite field. For example, over the field of two elements, the probability of determinant equal to zero is asymptotic to about 0.71.