I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending through the night into the morning. But I have never actually solved a problem in my sleep. I have awakened thinking That's it!, but never was it actually it.
Q. Can anyone report an actual significant advance in their research that occurred during and emerged from their sleep?
Of course this is entirely subjective, but you would know it if it happened to you.
Poincaré's famous step onto the bus in 1908 ("At the moment when I put my foot on the step the idea came to me...") indicates significant unconscious processing, and his insomnia account (quoted below) adds further credence to such "background" processing. But I am not aware of first-hand reports of significant and accurate reasoning occurring during sleep.
One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. ... (Link)