Advection driven problems are known to be difficult to model with a reduced basis because of a slow decay of the Kolmogorov $N$-width. This paper investigates how this challenge transfers to the context of solidification problems and tries to answer when and to what extend reduced order models (ROMs) work for solidification problems. In solidification problems, the challenge is not the advection per se, but rather a moving solidification front. This paper studies reduced spaces for 1D step functions that move in time, which can either be seen as advection of a quantity or as a moving solidification front. Furthermore, the reduced space of a 2D solidification test case is compared with the reduced space of an alloy solidification featuring a mushy zone. The results show that not only the PDE itself, but the smoothness of the solution is crucial for the decay of the singular values and thus the quality of a reduced space representation.