One common definition of a
*ruled surface*
$S$ is that, through *every* point $p \in S$, there passes a line $L(p)$ that lies
in $S$: $L(p) \subset S$.
My question is:

Q0. Is there any loosening of "through every point $p$" that leads to the conclusion that $S$ is nevertheless still ruled?

(*Added*:) Let us assume $S$ is smooth and $L(p)$ is an infinite line (not a segment).
More specifically:

**Q1**. Is there a surface $S$ through which all but a finite set of
exceptional points, passes a line that lies in $S$? So it is not ruled, but there are only finite exceptions?

**Q2**. Is there a surface $S$ through which all but a countably infinite set of exceptional points, passes a line that lies in $S$?

**Q3**. If a set of points $P$ is dense on $S$, and through every $p\in P$
there passes a line that lies in $S$, does that imply that $S$ ruled?

^{Image from Univ Arizona Math Teaching Tools}

(

*Added 14Oct14*). Here is Robert Bryant's example surface $S$ that shows the answer to

**Q1**is—remarkably—

*Yes*. From $z=xy$ he removes the four red points $$(1,0,0),\; (−1,0,0),\; (0,−1,0),\; (0,1,0)\;,$$ and then there is no complete line through the five yellow points: $$(0,0,0),\; (1,1,1),\; (1,−1,−1),\; (−1,1,−1),\; (−1,−1,1)\;,$$ but there is a line through every other point of $S$.

twopoints $(-1,1,-1)$ and $(1,-1,-1)$, every point of $S$ lies on a (complete) line in $S$. I don't know whether one can construct an example with only one exceptional point. $\endgroup$ – Robert Bryant Oct 14 '14 at 11:43