Recent observations of lensed galaxies at cosmological distances have detected individual stars that are extremely magnified when crossing the caustics of lensing clusters. In idealized cluster lenses with smooth mass distributions, two images of a star of radius R approaching a caustic brighten as and reach a peak magnification before merging on the critical curve. We show that a mass fraction ( ) in microlenses inevitably disrupts the smooth caustic into a network of corrugated microcaustics and produces light curves with numerous peaks. Using analytical calculations and numerical simulations, we derive the characteristic width of the network, caustic-crossing frequencies, and peak magnifications. For the lens parameters of a recent detection and a population of intracluster stars with , we find a source-plane width of for the caustic network, which spans on the image plane. A source star takes years to cross this width, with a total of crossings, each one lasting for with typical peak magnifications of . The exquisite sensitivity of caustic-crossing events to the granularity of the lens-mass distribution makes them ideal probes of dark matter components, such as compact halo objects and ultralight axion dark matter.