The answer is: It depends.
First of all you still have a reactance, i.e. every cable has an inductance which (mainly) depends on the geometry. Hence, the inductance of a superconducting wire is more or less the same as for e.g. a copper wire.
Now, there are two types of superconductors: Type I and type II.
All cables and commercial magnets are made from type II superconductors.
In a type II superconductor (e.g. all high-Tc superconductors) you have what is known as flux flow resistance where "magnetic" vortices (known as Abrikosov vortives) are moved around by the Lorentz force. Real cables are built so that they have a loot of "pinning centres" to trap these vortices.
As you can imagine this is a complicated topic and I am not an expert (I work with superconducting electronics, not power applications) but I think that sums it up.
JohnCU: There is no "skin depth" as such in superconductors. However, you have something known as the penetration depth which is somewhat similar.