This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ray of light) rolls inside, going in straight lines, and upon collision with the wall, the orbit is reflected.
It is intuitive that a statement like the following is true:
For almost every billiard, there exists an orbit that is dense everywhere inside it.
However, as far as I know this is still open. In fact, the last thing I heard was that it had just been proven for the case in which the billiard was an obtuse triangle with certain restrictions (but I have since forgotten the source, unfortunately).
Question: What is the current status of the problem?
Clarification The question is not about rectangular billiard tables, but in general about the balls rolling in more general shapes. 'Almost always' would then have to be given a meaning within the space of curves. (In fact, the problem is trivial in rectangles because an orbit with irrational slope will do.) Also, this is not about having dense families of orbits, but a single orbit that is dense in the billiard.
I think the way 'almost always' should be defined is by requiring some generic property to hold. Think, for example, of the definition R. Abraham give of bumpy metrics.