Ohm's Law does not have a problem here any more than any other formula in the sciences which involves dividing by a denominator which can go to zero.
Ohm's Law exhibits a singularity when there is no resistance, but a nonzero voltage. An ideal voltage source cannot be connected in parallel with a zero resistance, because that implies that infinite current flows, which is absurd.
Note that superconductivity doesn't eliminate impedance. Even if you had an ideal voltage source to connect across a piece of superconductor, the current would not be infinite. It would be limited by inductance (which would allow the current to gradually rise without bound). To properly model the circuit, it would have to be drawn as an ideal voltage source connected to an ideal inductor. Such a thing is mathematically possible and analyzable (and in fact probably occurs in numerous elementary textbooks as an example).
Ohm's Law is an idealization based on ideal resistance, which has no parasitic inductance or capacitance. As such, it breaks down long before we reach zero resistance. So the singularity at R=0 is purely academic. At R=0, we have a piece of wire which may be superconductive, but it exhibits capacitance and inductance.
Note, by the way, that in superconductors, current can flow without voltage. But this fits in with all ordinary laws that we apply in analyzing simple circuits. If draw the schematic of a ciruit which consists of a loop of ideal wire, then some finite current can flow in that loop forever without any potential differences anywhere in that loop. We can divide the loop in half, and each half can "think" that there is a current source in the other half.