Given that these things are neutral and effortlessly fly out of collapsing supernova cores, heavier elements and superconductors are not very likely to help.

By our current understanding, superconductors only exist under extreme cold, a condition never occurring naturally in stars and supernovas. So the fact that neutrinos fly out effortlessly does not preclude the possibility of a superconducting trap of neutrinos.

There is nothing about a superconductor that would trap a neutrino.

If new physics are allowed then it's simple. We just need a Neutrino detector that fits in a phone and doesn't need too much electricity.

That would require a phone made out of something other than matter.