Why are only singly and doubly ruled non-planar surfaces found? Why not triply ruled? Is there a reason why there are no triply-ruled surfaces found in spatial geometry?  Does it have to do with the fact that there are at most two dimensions/parameterizations for a surface?  If that's so, then do 3D hyper-surfaces in a 4D+ space allow for triply ruled surfaces?
To be more clear, I am trying to find a proof of sorts that can demonstrate this fact, and whether or not the proof can be generalized to higher dimensional surfaces and spaces.
 A: Well, one can show that the only doubly rules surfaces are the plane, the hyperbolic paraboloid, and the single-sheeted hyperboloid, and none of these is non-planar and triply-ruled.
A: One way to think about the fact that no surface in space can be triply ruled
as follows:
Firstly, one sees that a surface $S$ of degree $3$ or higher is not ruled at all; a cubic surface can contain at most $27$ lines in total, and higher degree surfaces typically contain no lines at all.  (One way to prove this is via an argument with incidence varieties; see the sketch in this tricki entry.)
So a non-planar ruled surface has to be a quadric (i.e. cut out by a degree $2$ equation).
On the other hand, if $\ell$ is a line passing lying on $S$, passing through a point $s \in S$, then $\ell$ lies in the tangent plane to $S$ at $s$.   Since $S$ is a quadric, when you intersect it with a plane, the intersection is a (possibly degenerate) conic section, and so can contain at most two lines.  Thus there are at most two lines on $S$ passing through any given point $s$, and hence a quadric is at most doubly ruled.
