Peaked or cusped solitary waves (peakons or cuspons) were introduced in 1993 by Camassa and Holm in their seminal paper on the exact theory of shallow water waves. We demonstrate that similar non-smooth solitons naturally appear from several different short-pulse equations in fiber optics. In this context, cusp formation is an universal feature that prevents ultrashort optical solitons from going beyond the critical nearly single-cycle duration.