Is it possible for the planets to align? We've all heard the statement that on the 21st of December, the planets in the solar system will "align" from the point of view of the Earth. I assume this means that they would all be in the same spot in the sky if we looked from here. The theory says that the alignment of the planets will somehow exert some influence on the Earth which would bring varying levels of catastrophe, depending on who you ask.
Now, it has been said many times that this will not actually happen, and that even if it happened there would be no effect on the Earth whatsoever. I know that, and that's not the question.
What I'm wondering here if it is actually possible for the planets to align in this way, regardless of whether it'll actually happen. As far as I know, the planets' orbits aren't all in the same plane, so it doesn't seem even theoretically possible, i.e., there's no straight line passing through the orbits of all the planets. Am I right?
 A: First, Mercury "aligns" with the ecliptic plane only twice in its "year", when it comes from above to below and vice versa.
Luckily for our calculations, Pluto is not a planet any longer, because it would completely rain on our parade with its 248 Earth years of orbital period and another two points within it that it crosses the plane again. Getting Pluto and Mercury aligned alone would take millennia.
Now, what do we count as "aligned"? This is a very vague term because it doesn't state any tolerances. If you mean discs of the planets overlapping, just forget it, their own minor deviations from the ecliptic plane will suffice that it will never ever happen. Let us assume a tolerance of one earth day of their movement. This is fairly generous, in case of Mercury it's over 4% tolerance of its total orbit radius, which considering their size on the sky is quite a lot - in case of all planets the distance traveled over one earth day far exceeds their diameter. So, we're not taking a total alignment, just one night where they are closest to each other, a pretty loose approximation.
Now, we pick the day the rest of the planets are on the plane as Mercury, so let us simply take the 2 in 88 days of its orbital period and continue dividing by orbital periods of other planets.
1 in (44 * 225 * 365 * 687 * 4332 * 10759 * 30799 * 60190) days.
That is one day in $5.8 \cdot10^{23}$ years.
The age of the universe is $1.375 \cdot 10^{10}$ years.
It means planets would align for one day in 42 trillion times the age of the universe.
I think it's a good enough approximation to say it is not possible, period.
Feel free to divide by 365, if you don't want aligned with the Sun but only with Earth. (one constraint removed.) It really doesn't change the conclusion.
A: All the planets except Mercury (7 degrees off) and Pluto (17 degrees off) are on the ecliptic plane. So a perfect alignment is not possible. I'm including Pluto as a planet out of habit.
A: Theoretically, no, as the plane of orbit of each planet is tilted slightly in relation to other planets. However, if we ignore the intra-orbital plane tilt, then the chance of all planets (eight now excluding Pluto) to be at the right ascension with respect to sun at same time would be once in 180 trillion years. 
For an exact alignment, when all the planets are inclined with respect to the ecliptic, we must factor the line-of-nodes recession into calculations, that makes the chance once in 86 billion trillion trillion trillion years (86 followed by 45 zeros). 
The odds strongly favour the fact that an exact planetary alignment will never occur within the lifetime of the solar system that now has only about 12 billion years left. 
— S P S Jain, Greater Noida 
A: The orbital planes are all different. However, the orbital planes do intersect, and the orientation of the orbital planes precesses slowly. Therefore, it is mathematically possible that at some moment $t$, all orbital plane intersections would be at the same angle and all planets would be at this position within their orbital plane. One could do the calculations, but I'd expect that this state is so unusual that the expected time to wait for it is longer than the expected lifetime of the universe.
