We use fast confocal laser microscopy to quantitatively study axisymmetric and nonaxisymmetric shapes of lipid vesicles undergoing spontaneous transformations. To characterize the observed three-dimensional shapes, we compute their respective reduced monolayer area difference and reduced volume. The transformations allow us to analyze a broad range of vesicle shapes including stomatocytes, elliptocytes, discocytes, cigars, necklaces, and many nonaxisymmetric shapes. The transformations are marked by a step-like time dependence of the reduced monolayer area difference, which is explained in terms of the area-difference-elasticity model. The simplest mechanism consistent with the observed shape transformations are small folded multilamellar patches on either monolayer. The experimental methodology validated by the reported results can be used to monitor and analyze more complex vesicle shapes and their evolution.