Is this an example of spontaneous symmetry breaking?

Consider a pencil standing in a (ideally) perfectly vertical position.

The gravitational field will the same no matter the (angular) direction it will fall in.

But it will end up falling in a particular direction.

Is this an example of spontaneous symmetry breaking? Or is just that due to quantum fluctuations the pencil will never actually be perfectly vertical?

Answer

It is indeed a an example of symmetry breaking. Basically, the problem (the equactions, and the fields, aven the solution) is symmetric, however because the solution is unstable, in a realistic case where noise is present will break the symmetry into a solution that is not symmetric. The solution "choose" one specific direction to get to a state of lower energy, and because of this choice, the solution is not symmetric anymore. In one of the comments you read

Spontaneous symmetry breaking is a technical term almost always connected to a phase transition/order parameter

which is correct, but not necessarily so. In this case the symmetry breaking is the choice one one particular ground state (the pencil resting horizontal on the table) among equaly probable others of the same energy.

From wikipedia:

Spontaneous symmetry breaking is a mode of realization of symmetry breaking in a physical system, where the underlying laws are invariant under a symmetry transformation, but the system as a whole changes under such transformations, in contrast to explicit symmetry breaking. It is a spontaneous process by which a system in a symmetrical state ends up in an asymmetrical state. It thus describes systems where the equations of motion or the Lagrangian obey certain symmetries, but the lowest-energy solutions do not exhibit that symmetry.

The example asked here staisfies that the equations of motion or the Lagrangian obey certain symmetries, but the lowest-energy solutions do not exhibit that symmetry. As such, it has to be defined as a speontaneous symmetry breaking.

On the other hand, explicit symmetry breaking is the breaking of a symmetry of a theory by terms in its defining equations of motion (most typically, to the Lagrangian or the Hamiltonian) that do not respect the symmetry.