Which way will the pencil fall? Let's say you had a perfect pencil, with a point which was just that one point (see this question). The pencil's mass was perfectly distributed, and there are no flaws in the craftsmanship. Let's say you had it oriented vertically (like you were going to balance it), and it was exactly straight up and down i.e. the pencil's center of mass is directly above the point where the pencil is touching the ground. In the universe where this pencil is, there are no outside forces which can affect the pencil, other than gravity.

Which way will the pencil fall, after you let go?

Is it random?
 A: 
Which way will the pencil fall, after you let go?

Facetious answer: it would fall in the direction to which it was leaning when you let go.
Okay, now to justify that: you cannot balance the pencil before letting go. For an object resting on a surface to be balanced, its centre of mass (a single point) must be directly above a point within the area of contact between it and the surface beneath.
In your scenario the area of contact is a single point (i.e. the surface area is zero), so it is impossible to find a point within this area above which to position the pencil's centre of mass and balance it. There will always be a line you can draw between the point of contact and the point on the surface below the pencil's centre of mass - the pencil will fall along this line.
A: 
In the universe where this pencil is, there are no outside forces which can affect the pencil, other than gravity.

But there are forces afoot inside the pencil.
Unless you also chill the pencil to absolute zero and thus stop all molecular activity, the trillions of atoms in the pencil are vibrating in random directions. It won't take long for the random force vectors to push the pencil's center of mass outside the column of stability, which will result in a tiny amount of torque, which will quickly cascade until the pencil is lying on it's side.

[control of random molecular movement] were often used to break the ice at parties by making all the molecules in the hostess's undergarments leap simultaneously one foot to the left, in accordance with the theory of indeterminacy.

A: What you've fabricated is of course unrealistic in the physical world as CuriousOne stated, but not so in the virtual world of simulation. All the conditions you ask for can be arranged in a simulated universe. If perfectly balanced as its initial condition, the virtual pencil will not fall in this virtual world. It is unstable, but it will not fall until a perturbing force, however small knocks it off balance. It will then fall in the direction of that perturbing force. 
A: I'll look at the question from multiple aspects.
Classical mechanics, exact measurements, no thermodynamics, no perturbing forces 
In this fictitious universe it is possible to stand our perfectly balanced pencil exactly vertically and perfectly stationary. This is an unstable equilibrium position.
With no perturbing forces, no thermodynamics, no quantum mechanics, the pencil won't fall.

Classical mechanics, exact measurements, no thermodynamics, no perturbing forces except tides
You can balance the pencil vertically for an instant. A moment later, the tidal forces from the Moon and the Sun will have shifted the direction of local vertical. (Aside: tides are gravitational.)
The pencil will fall away from the direction in which local vertical is moving.

Classical mechanics, exact measurements, classical thermodynamics
The pencil will fall in some random direction.

Classical mechanics, realistic measurements
In theory, one can measure to infinite precision in classical mechanics. In practice, we cannot do that. In the world of pencils and bridges, the engineering limitations on ability to measure precisely swamp the tiny quantum mechanical errors inherent in measuring conjugate variables.
The pencil will fall in some random direction.
Comment: This is an inverted pendulum. They fall over -- unless something acts to move the inverted pendulum back toward the unstable equilibrium. Thousands of mechanical engineering students face this problem every year.  They have to construct a robot that keeps an inverted pendulum inverted.

Quantum mechanics
The question is tagged uncertainty-principle. The only way to have the pencil be in this unstable equilibrium state is to have perfect simultaneous measurements of its orientation and angular momentum (or equivalently, the position of its center of mass position and its linear momentum). These are conjugate variables. The uncertainty principle says the product of the uncertainty of a pair of conjugate variables is at least $\hbar/2$.
This question doesn't make sense in a quantum mechanics context; it is akin to asking what the laws of physics says will happen given a violation of the laws of physics.
A: Suppose that we align a perfectly cylindrical pencil with the z-axis. If the initial conditions are rotationally symmetric about that axis, then because the laws of physics are rotationally symmetric, the final state must also be symmetric under rotations about the z-axis. This means that the wavefunction of the universe will have to evolve into a superposition of the pencil falling in every possible direction with the rest of the universe being entangled with these possible outcomes.
