The generation of dark spin wave envelope soliton trains from a continuous wave input signal due to spontaneous modulational instability has been observed for the first time. The dark soliton trains were formed from high dispersion dipole-exchange spin waves propagated in a thin yttrium iron garnet film with pinned surface spins at frequencies situated near the dipole gaps in the dipole-exchange spin wave spectrum. Dark and bright soliton trains were generated for one and the same film through placement of the input carrier frequency in regions of negative and positive dispersion, respectively. Two unreported effects in soliton dynamics, hysteresis and period doubling, were also observed.