The nuclear envelope (NE) presents a physical boundary between the cytoplasm and the nucleoplasm, sandwiched in between two highly active systems inside the cell: cytoskeleton and chromatin. NE defines the shape and size of the cell nucleus, which increases during the cell cycle, accommodating for chromosome decondensation followed by genome duplication. In this work, we study nuclear shape fluctuations at short time scales of seconds in human cells. Using spinning disk confocal microscopy, we observe fast fluctuations of the NE, visualized by fluorescently labeled lamin A, and of the chromatin globule surface (CGS) underneath the NE, visualized by fluorescently labeled histone H2B. Our findings reveal that fluctuation amplitudes of both CGS and NE monotonously decrease during the cell cycle, serving as a reliable cell cycle stage indicator. Remarkably, we find that, while CGS and NE typically fluctuate in phase, they do exhibit localized regions of out-of-phase motion, which lead to separation of NE and CGS. To explore the mechanism behind these shape fluctuations, we use biochemical perturbations. We find the shape fluctuations of CGS and NE to be both thermally and actively driven, the latter caused by forces from chromatin and cytoskeleton. Such undulations might affect gene regulation as well as contribute to the anomalously high rates of nuclear transport by, e.g., stirring of molecules next to NE, or increasing flux of molecules through the nuclear pores.