Quasicrystals and their periodic approximants are complex crystalline phases. They have now been observed in many metallic alloys, soft matter systems, and particle simulations. In recent experiments of thin-film perovskites on solid substrates, the type of complex phase was found to change depending on thermodynamic conditions and the type of substrate used. Here, we investigate the effect of a substrate on the relative thermodynamic stability of a two-dimensional model quasicrystal and its approximants. Our simulation model is particles interacting via the Lennard-Jones–Gauss potential. Our numerical methods are molecular dynamics simulations and free energy calculations that take into account phason flips explicitly. For substrates interacting weakly with the particles, we observe an incommensurate–commensurate transition, in which a continuous series of quasicrystal approximants locks into a small number of approximants. Interestingly, we observe that the 3/2 approximant exhibits phason mode fluctuations in thermodynamic equilibrium. Such fluctuations are reminiscent of random tiling and a phenomenon usually associated only with quasiperiodic order. For stronger substrates, we find an enhancement of the stability of the dodecagonal quasicrystal and variants of square lattices. We explain all observed phenomena by the interplay of the model system with the substrate. Our results demonstrate that designing novel complex periodic and quasiperiodic structures by choice of suitable substrates is a promising strategy.