Subsystem symmetries act on rigid subextensive subsystems, i.e. lines, planes, or fractals, and have recently gained interest due to their connection to fracton topological order in 3D. Such symmetries can also protect non-trivial symmetry protected topological (SPT) phases. In 2D, these can be regarded as "symmetry-protected fracton" phases without topological order. We show how one can make sense of and classify all possible phases in 2D for two general types of subsystem symmetries: linear and fractal subsystem symmetries with a finite Abelian on-site symmetry group.