It has been shown analytically that Peregrine solitons emerge locally from a universal mechanism in the so-called semiclassical limit of the one-dimensional focusing nonlinear Schrödinger equation. Experimentally, this limit corresponds to the strongly nonlinear regime where the dispersion is much weaker than nonlinearity at initial time. We review here evidences of this phenomenon obtained on different experimental platforms. In particular, the spontaneous emergence of coherent structures exhibiting locally the Peregrine soliton behavior has been demonstrated in optical fiber experiments involving either single pulse or partially coherent waves. We also review theoretical and numerical results showing the link between this phenomenon and the emergence of heavy-tailed statistics (rogue waves).