Dense orbits in billiards 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?

Thank you!
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.
 A: Do you mean to ask whether the trajectories in almost all cases (in {shapes X trajectories}
are dense in the set of {positions, directions} on the table, or just in the set positions?
The first question seems more natural to me; the answer is no: 
If there are
two convex portions of the boundary curve pointing toward each other, they're like 
convex mirrors, they tend to focus. For open sets of shapes and distances, the return
map from the tangent line bundle along a mirror back to itself has eigenvalues
of the first derivative
a pair of complex conjugate points on the unit circle.  Because of the KAM theory,
there are typically rings of positive measure consisting of invariant circles for the
return map. The orbits of these rings under the billiard flow enclose an open set in phase space.  
Another physical example of this effect what happens when you wind something
like kite string
around a flat object, perhaps a piece of cardboard or a board. The string tends to
build up in the middle, and once a bulge gets started, the configuration is stable---the
string prefers to wind back and forth across the bulge. (Note that the shortest paths of
winding string follow geodesics, which are the same as trajectories as billiards
on a table of the same shape).  
Even when the KAM situation isn't obvious from the geometry of the table, 
I think experimental evidence shows that it's commonplace. There are known constructions
of Riemannian metrics on $S^2$ with ergodic geodesic flow, but they took a long
time before someone found them (sorry, I don't remember the reference).  Similarly,
I think it's tricky to find examples of simply-connected billiard tables with smooth boundary that are ergodic: you somehow have to systematically eliminate the KAM phenomenon.
It's much easier if the table either has angles, or is multiply-connected with two 
or more obstacles in the middle (so that doubling it produces a surface of negative
Euler characteristic).
It's not obvious to me how to use this phenomenon to 
capture all trajectories that pass through a particular point, but maybe that's
not really the most natural question: after all, in a game of billiards, direction and position both matter.
A: Assuming that the OP is about billiard tables, not strangely shaped billiard balls:
The result for (some) obtuse triangles is the opposite of what the OP wants: Rich Schwartz has proved that (some) obtuse-angled triangles have periodic orbits, see:
MR2549685 (2010g:37060) 
Schwartz, Richard Evan(1-BRN)
Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. (English summary) 
Experiment. Math. 18 (2009), no. 2, 137–171. 
That in polygonal billiards with angles rational multiples of $\pi$ the generic orbit is dense follows from the classic result
MR0855297 (88f:58122) 
Kerckhoff, Steven(1-STF); Masur, Howard(1-ILCC); Smillie, John(1-CUNY7)
Ergodicity of billiard flows and quadratic differentials. 
Ann. of Math. (2) 124 (1986), no. 2, 293–311. 
58F17 (30F30 58F11) 
See the following for more along the lines of specifically dense orbits:
MR1458298 (98k:58179) 
Boshernitzan, M.(1-RICE); Galperin, G.(D-BLF-BB); Krüger, T.(D-BLF-BB); Troubetzkoy, S.(D-BLF-BB)
Periodic billiard orbits are dense in rational polygons. (English summary) 
Trans. Amer. Math. Soc. 350 (1998), no. 9, 3523–3535. 
58F20 
Serge Tabachnikov's book "Geometry and Billiards" is highly recommended.
A: It is possible for a billiard path to be dense and yet not be uniformly distributed. This is true for the triangle with angles $0.4\pi,0.3\pi,0.3\pi$. (See, for example, theorem 1.3 of http://homepages.math.uic.edu/~demarco/billiards.pdf.)
