Final answer:
The nuclear membrane disappears during the prophase stage of mitosis, allowing spindle fibers to attach to kinetochores on sister chromatids.
Step-by-step explanation:
The nuclear membrane disappears during the prophase stage of cell division, specifically mitosis. During this phase, the nuclear envelope breaks down into small vesicles, and cellular organelles like the Golgi apparatus and the endoplasmic reticulum also disperse.
The nucleolus vanishes, centrosomes migrate to the cell poles, spindle fibers form, and the chromosomes condense and become visible. The complete disintegration of the nuclear envelope allows the spindle fibers to connect with the kinetochores on the sister chromatids, preparing them for segregation to opposite poles.