This is partly why high temperature superconductors aren't actually all that useful (the current capacity at LN2 temperatures stinks), and why LHe is usually used with most superconducting magnets.
And also why for lossless transmission, perfect conductors would be much better than superconductors.
Incidentally, for AC transmission in particular, superconductors are... weird.
For one, cable is usually superconductor embedded in copper -- it's stronger, it's resistive at higher temperatures, and it provides some bulk dissipation capacity in the event of a quench.
Copper becomes an ever-better conductor at low temperatures, but never a superconductor: it has residual resistance. This means all the usual AC loss effects are still in play, even if the cable is mostly superconducting. The copper has some skin effect (which, at low temperatures, is fairly shallow, even at mains frequency, due to its improved conductivity), so the superconducting strands need to be placed towards the surface (which explains the cross-sections shown in typical drawings of this stuff).
You can also make cables out of the superconductor alone, to varying degrees of "can"; most of these materials are very brittle, so it can be difficult to manufacture and handle. If executed correctly, you get a high Q factor, for basically all frequencies from DC to "light".
Type 1 superconductors are great at this; niobium resonators are used in LINACs at ~1000MHz, with Q factor in the 10^7 range. The Q is decidedly finite, though, and the loss mechanisms are not understood. It depends greatly on surface finish.
Type 2 superconductors are noticeably bad. In fact this can be
felt: a magnet can be pushed into such a superconductor, where it will stick in place (flux pinning). A force was applied, over some distance: work! It's lossy! In fact, it exhibits hysteresis loss. We should expect (I haven't seen any numbers on this unfortunately -- but it should be the case) that this gives a much lower Q for AC currents.
There are different forms of conductor as well. Thin films can be made in very good quality. AFAIK, this is what's being used for quantum computing devices. AFAIK, the crummy ones are polycrystalline, so flux pinning probably has something to do with grain boundaries. Maybe thin films can be deposited on carrier wires, and those embedded in cable for power? Dunno.
Tim