A characterization of the return-difference matrix properties of optimal linear stationary estimation in singular mode is presented in which the non-singular Kalman problem and the coloured measurement noise problem occur as special cases. Linear least-squares estimators of minimal order are treated for both known state-space models and known separable output covariance models. Return-difference properties of the optimal linear regulator, where the control weighting matrix is singular, are also shown to be the dual of those of the singular estimation solutions.