In this note we study the disturbance decoupling problem for singular systems E(x) over dot = Ax + Bu + Gd, y = Cx, x = Hx with left invertibility, i.e., [(aE-A)(H) (B)(0)] is left invertible. We give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability by output feedback that also makes the resulting closed-loop system regular and/or of index at most one. All results are proved based on a condensed form that can be computed using orthogonal transformations, i.e., can be computed in a numerically stable way.