We present two single-shot experiments devoted to the study of noise-induced modulation instability (MI) in optical fibers. The first one, performed in a recirculating fiber loop, enables the first observation of the space-time dynamics of MI. The second one uses a heterodyne version of time microscope (SEAHORSE) that enables the single-shot observation of the phase and of the amplitude of the nonlinear random waves. Our SEAHORSE has a subpicosecond resolution and allows observation within a window of observation of 160ps. Our observations provide new experimental insights into the fundamental problem of spontaneous MI.