Such basic information seems necessary to engineer any purpose-built device utilizing superconductors, yet the only papers I'm seeing are geared towards niche roles in quantum computing. These seem to hint of continued parasitic losses, but little else.
Inductive impedance results from magnetic field energy storage, and superconductors have a nearly-complete exclusion of magnetic field in their bulk, so there's a bit of an inductance drop when a material becomes superconducting, and its skin depth for any alternating current becomes nil.
As for capacitance, that results from electric field energy storage, and is dominated by external fields around both conductors and superconductors; there's no reason to expect change there.
An important reason to care about capacitance and inductance is the transmission line effect (delay of propagating signals) and that is slightly complicated by the fact that superconductors are less than 'super' when conducting alternating currents; losses occur, and hysteresis (which makes a phase delay). The transmission-line properties of a superconductor are... interesting.
J. F. Whitaker, R. Sobolewski, D. R. Dykaar, T. Y. Hsiang and G. A. Mourou, "Propagation model for ultrafast signals on superconducting dispersive striplines," in IEEE Transactions on Microwave Theory and Techniques, vol. 36, no. 2, pp. 277-285, Feb. 1988, doi: 10.1109/22.3516.
There is still inductance in superconductors. In fact, with a resistance of zero the inductance becomes very important. You also have mutual inductance to nearby conductors (which may or may not be superconducting). In the latter case, there can be losses from any AC component in the supercurrent.
Stray capacitance similarly will exist as with any conductor.
Nothing is ideal, so you can stop worrying about whether something is or not. You can still worry about how close something is to ideal, for your purposes.
You'll still have transmission line effects, so a cap made from superconducting plates will still have inductance and a self-resonant frequency and whatnot. Similarly, a wound inductor will still have mutual capacitance between the coils.
If you have a resonant tank circuit made out of superconducting materials in free space, resonating away, then there will be coupling to free space and thus radiation loss. I can't be sure, but I suspect that even if you put your superconducting tank circuit into a superconducting shield that energy would find some way of getting out -- and if it never ever could get out, you wouldn't have a device that's useful to anyone.
Capacitance and Inductance are names given to certain properties of conductor geometry.
If you don't alter the geometry, then L and C don't change.
As superconductors tend to expel current from their interior and conduct through their skins, there is a very minor change in effective conductor geometry but nothing serious.
