Is there any exception to the rule of non-contradiction? In physics, mathematics or philosophy? Is there any system where the law of non-contradiction doesn't apply for a good reason? I can't think of any. I am wondering if quantum superposition is an exception to this rule.
There are no real-world examples, because real things don't have opposites, so long as we don't treat the lack of a property as itself a property.
The principle of non-contradiction is mostly a linguistic tool, to point out the absurdity in saying "a non-married married person," for example. Quantum physics also does not contradict this, because superposition, for example, means that every quantum state can be represented as the sum of other quantum states. At worst, this means that two different states can be true at the same time - which is not a contradiction, but rather an ability for particles to have more than one property in areas we normally think of as having only one property.
For example, if a particle is in position A and also in position B, this doesn't mean it is in the position A and also the position not-A, because "not" is a logical function, not the negation of a property; there is no position that is the logical opposite of position A. You can think of this through the analogy of a zebra, which has the property of being white, and also being black, but "black" and "white" are not logical opposites, so there is no contradiction.
The example you are likely thinking of, Schrodinger's Cat, was not intended as a demonstration of quantum physics, but rather the absurdity of the Copenhagen Interpretation.