Capacitor in a superconducting circuit Assume we have started a current in a closed circuit which only includes a capacitor. No inductor.
For example by having a battery initially, disconnecting and and reclosing the circuit.
For a dissipative conductor, the current would decrease with the timescale of the inverse scattering  rate.
What if the circuit were superconducting? Would there be an oscillation of the current with charge sloshing back and forth each plate of the capacitor? At which frequency? 
 A: To take your question at face value: what if we somehow connected the two ends of a charged capacitor with an ideal wire, that somehow avoids having self-inductance or radiation losses or anything?
Well, we can write a differential equation using Kirchoff's voltage law: $$ \frac{Q}{C}=0 $$
Whoops! That's nonsense- our initial conditions are not $C=\infty $ or $Q=0$, so we have a contradiction. Which means our assumptions are bad. Basically the problem we run into is that you can ignore the inductance of a wire when the circuit is dominated by other effects, but the more ideal you make your circuit components, the more whatever inevitable non-ideal properties the other components have stand out.
In particular, any loop of wire necessarily has a self-inductance. If the self-inductance is $L $, ignoring radiation losses, then our equation becomes the much more sensible  $$L\ddot{Q}+\frac{Q}{C}=0$$
which gives the solution $Q(t)=Q_0\cos(\omega t) $, where $\omega=\frac{1}{\sqrt{LC}}$.
For small radiation losses, this formula is still approximately correct if $Q_0$ is replaced with a slowly varying function of time. 
So, yes, in the limit that the radiation losses are small, the charge does slosh back and forth, with a frequency determined by the size of the inductance.
As John mentions in his answer, this is by no means the only possibility. While you can never decrease the inductance to zero, you can make it small enough that some other effect takes over.
A: I have to disagree with Chris and say there is no reason to require a non-zero inductance to be present (though of course in practice there will always be a non-zero self-inductance). Electrical signals do not propagate instantaneously, so it's perfectly possible to get a stable oscillation even with zero inductance. Your system will oscillate at the plasma frequency of the conductor.
However the capacitor is not really playing any special part in this. Any chunk of metal can sustain plasma oscillations as they are a fundamental property of any electron gas.
