I apologize if this is too much of a fishing expedition, but I've had bad luck searching for any literature on this subject, and I was hoping someone could tell me if it's too easy to be worth mentioning, too hard to say anything about, or if I'm using the wrong key words.
Let $X\to Y$ be a resolution of singularities.
Is there any adjective I could apply to the above resolution which would assure that its scheme-theoretic fibers are reduced? Perhaps crepant, or maybe symplectic? What if the base has rational singularities?