Is there any particular problem or scenario where quantum logic may be applied? If so, what is the benefit of using quantum logic instead of classical logic?
I've been reading quite a lot on this topic but it still seems like pure theoretic considerations to me. If there is so much effort put into forming new type of logic based on quantum mechanics, are there any practical uses of this type of logic?
The Quantum Logic Explorer (Theorem List) and the Hilbert Space Explorer of the Matamath project give a good overview of the type of theorems proved with quantum logic. I would say that all theorems stay on the level of propositional logic, which doesn't feel too useful to me. Classical logic becomes useful and interesting in the context of first order predicate logic. As long as quantum logic has no comparable predicate logic, it probably won't be (too) useful. A compromise might be a modal quantum logic, but I can see no trace of such a logic either.
One reason why modal or predicate quantum logics are not developed yet is that even the propositional quantum logic still hasn't succeeded in really capturing the propositional logic of the closed subspaces of a Hilbert space, as Gian Carlo Rota sometimes stressed. The work by David Ellerman on quantum logic is probably crucial to bring at least the propositional quantum logic to an acceptable state (where it appropriately captures the logic of subspaces), but even this is not yet fully worked out, and still contains no traces of a modal or predicate logic.
A completely different approach which is also sometimes called quantum logic develops a category theoretical framework for Hilbert spaces (dagger categories), and presents connections to related parts of mathematics and (linear) logic, like the Rosetta Stone paper by John Baez and Michael Stay. This captures at least some aspects of quantum mechanics quite well, like the time reversibility and the "no cloning" theorems. But it seems to ignore the time dynamics of quantum mechanics, even if it is not alone with this shortcoming. Most text on quantum computing and quantum information also don't talk much about the actual time dynamics.