In this paper we extend a classical result, namely, the one that
states that the only doubly ruled surfaces in $\mathbb R^3$ are the
hyperbolic paraboloid and the hyperboloid of one sheet, in three
directions: for all space forms,
for any dimensions of the rulings and manifold, and to the conformal
realm. We show that all this can be reduced, with the help of quite
natural constructions, to just one simple example, the rank one real
matrices. We also give the affine classification in Euclidean~space.
To deal with the conformal case, we make use of recent developments
on Ribaucour transformations.