We investigate the structure of the quasiparticle states localized in a superconducting vortex core in the vicinity of a planar defect. It is shown that even a highly transparent defect leads to a significant modification of the excitation spectrum, with the opening of a minigap at the Fermi energy. The magnitude of the minigap exceeds the mean level spacing for a clean vortex already for small values of the reflection coefficient off the defect. It is maximal for the vortex sitting right at the defect, decreases with increasing the distance from the defect and closes at some point. We then generalize the problem for various configurations of several linear defects (periodic structures, two crossing lines, stars). Though the minigap remains, a strong commensurability effect is observed. For two crossing linear defects, the magnitude of the minigap strongly depends on how close the intersection angle is to a rational number.