"Singular values" are just the eigenvalues of the matrix.

This isn't accurate: the singular values of a matrix (in general of an operator in a Hilbert space that blah-blah) are the eigenvalues of the normal matrix (operator) which can be very different from the eigenvalues of itself. That the determinant of any matrix is the product of its eigenvalues (in some field extension of the definition field) is basic stuff in linear algebra. Singular values, imo, aren't and, besides, belong a little more to functional analysis than to just linear algebra. Tonio
Any matrix is similar to a matrix in either diagonal form or Jordan normal form where the determinant is just the product of the values on the diagonal. And the values on the diagonal are the eigenvalues.

And if a matrix has any 0 eigenvalue, then its determinant

**is** 0.

(If the determinant were the product of the "non-zero" singular values, then NO matrix would have 0 determinant!)