Superconductors are in the name. They're ideal conductors. As a corollary, superconductors coiled into a wire, shaped like an inductor, are also ideal inductors. By ideal, I informally mean nearly perfect in a mathematical sense.
A thin superconductor can carry an almost arbitrary amount of electricity with 0% loss. This is a real-world application already for superconductors, but it requires cooling the entire conductor to liquid helium temperature. (It's not truly limitless - enough current will eventually break down the superconducting effect - but ten billion watts down a 1 mm thick wire is doable.)
Similarly, an inductor made out of a superconductor, that is looped back on itself, can hold a magnetic field indefinitely, with 0% loss. Energy storage.
Also, novel ways of manipulating magnetic fields, and as a consequence of that, novel ways of manipulating radiation that interacts with magnetic fields. Really, anything that needs a strong magnetic field could benefit. Maglev trains. Portable MRI scanners would exist today, if the electromagnet didn't need to be submerged in liquid helium.
Superconducting computer circuits would dissipate no heat other than for the work required to physically change the state of the transistors. Power consumption could decrease by several orders of magnitude. Though to be honest, one day printing room-temperature superconductors lithographically is a rather unlikely prospect. But one can hope.
And some proposed realizations of quantum computing would benefit from small, extremely powerful magnets, while other proposed methods exploit the properties of superconductors directly (Josephson effect).