The quasi-2D platinum-based rare earth intermetallic LaPt2Si2 has attracted attention as it exhibits strong interplay between charge density wave order and superconductivity. However, most of the results reported on this material come from theoretical calculations, preliminary bulk investigations and powder samples, which makes it difficult to uniquely determine the temperature evolution of its crystal structure and, consequently, of its charge density wave transition. Therefore, the published literature around LaPt2Si2 is often controversial. Here, by means of high-resolution synchrotron X-ray diffraction data, we clarify some of the poorly or partially understood aspects of the physics of LaPt2Si2. In particular, we resolve the complex evolution of its crystal structure and superstructures, identifying the temperature dependence of multiple density wave transitions in good quality LaPt2Si2 single crystals. According to our findings, on cooling from room temperature LaPt2Si2 undergoes a series of subtle structural transitions which can be summarised as follows: second order commensurate tetragonal (P4/n m m)-to-incommensurate structure followed by a first order incommensurate-to-commensurate orthorhombic (P m m n) transition and then a first order commensurate orthorhombic (P m m n)-to-commensurate tetragonal (P4/n m m). The structural transitions are accompanied by both incommensurate and commensurate superstructural distortions of the lattice. The observed behavior is compatible with discommensuration of the CDW in this material.