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 $S$.