 The experimental observation of transverse modulations in proton beams accelerated from micrometre-thick targets irradiated with ultrashort laser pulses of a peak intensity of 5 by 1020 Wcm-2 suggests that the onset and strength of these modulations depend on laser energy and target thickness, and are likely caused by intensity-dependent instabilities in the laser-produced plasma at the target front side leading to electron beam break-up or filamentation. This article was authored by J. Metzges, T. Kluge, K. Zale, and others.