What actually oscillates in quantum harmonic oscillator with given frequency? Quantum harmonic oscillator is said to be describing motion of microscopic stuff (like atoms in molecules). But unless one keeps on measuring the position on the atom, it doesn't exist at all. It's in superposition of several possible positions (position eigenstates). How can we even assume oscillation and specifically with given frequency of oscillation?
 A: Even though you can't talk anymore about definite values of coordinates (and of other observables) you still can talk about quantum state and probability distributions it corresponds to. Their evolution is very specific determined by the Hamiltonian. We could call this system "quantum oscillator" simply because of its classical limit without reference to anything classical in the quantum regime.
However there are still good reasons to call this system "oscillator". If you look on the coordinate and momentum operator in the Heisenberg picture (i.e. time-dependent operators acting on the initial state) you'll discover that they satisfy Heisenberg equations of motion,
$$
\frac{d}{dt}\hat{x}(t)=\frac{1}{i\hbar}[\hat{x}(t),\hat{H}]=\hat{p}(t),\quad
\frac{d}{dt}\hat{p}(t)=\frac{1}{i\hbar}[\hat{p}(t),\hat{H}]=-\omega^2\hat{x}(t)
$$
from which we can derive e.g. that,
$$
\hat{x}(t)=\hat{x}(0)\cos{\omega t}+\hat{p}(0)\sin{\omega t}
$$
So it's observables in the Heisenberg picture that oscillate!
That translates to the evolution of the expectation values and probability distributions. We can derive oscillatory behaviour for the expectation values. For probability distribution the general conclusion is that no matter what initial state we consider they will be the same if we consider them with time shift $\delta t=2\pi/\omega$ and they also very simply related for $\delta t=n\pi/2\omega$. What happens in between mostly depends on the state. The most trivial case is simply stationary state.
Other interesting case are so called coherent states special for harmoning oscillator. They are stable wavepackets that follow classical trajectory without dispersing out. You can represent all other states as their superpositions.

A: First of all, I would agree that the particle "is in superposition of several possible positions (position eigenstates)" but I would strongly disagree that "it doesn't exist at all."  But anyway: if the quantum harmonic oscillator is in an energy eigenstate, then nothing oscillates at all; this is true for any energy eigenstate of any time-independent Hamiltonian.  The phase factors $e^{-iEt}$ from solving Schrodinger's equation always cancel.  But if you're not in an energy eigenstate, then the expectation value $\langle \psi | \hat{X} | \psi \rangle$ does indeed oscillate in time with a frequency of $\omega$, just as with a classical harmonic oscillator.  This is a special case of Ehrenfest's theorem.
A: Hamiltonian eigenstates viz. $\hat{H}\left|n\right\rangle =E_n\left|n\right\rangle$ span the Hilbert space. At time $t$ the state vector may be written as $\left|\psi\left( t\right)\right\rangle =\sum_n \psi_n\left( 0\right)e^{-i\omega_n t}\left|n\right\rangle$ with $\omega_n=\frac{E_n}{\hbar},\,\psi_n\left( 0\right)=\left\langle n|\psi\left( 0\right)\right\rangle$. The $e^{-i\omega_n t}$ factors are complex phases. It is the relative phases of energy eigenstates of different eigenenergies that oscillates (though as other answers have noted, certain other things do as well).
