Any problem where a statement can have a range of truth values, such as fuzzy logic. Consider: it is raining. You might want to assign a truth value to this statement that has more resolution than simply true or false: raining cats and dogs, steadily raining, sprinkling, misting, no precipitation.
However, the law of the excluded middle is perfectly fine, in my mind, when all the statements in question have only true or false values.
Doubtless emakarov can produce a more sophisticated answer than this, but these are my two cents.