  We employ the microcanonical inflection-point analysis method, developed for
the systematic identification and classification of phase transitions in
systems of any size, to study the two-dimensional Ising model at various
lattice sizes and in the thermodynamic limit. Exact results for the density of
states, which were obtained by exact algorithmic computation, provide evidence
for higher-order transitions in addition to the well-studied second-order
ferromagnetic-paramagnetic phase transition. An independent third-order phase
transition is identified in the ferromagnetic phase, whereas another
third-order transition resides in the paramagnetic phase. The latter is a
dependent transition, i.e., it is inevitably associated with the critical
transition, but it remains separate from the critical point in the
thermodynamic limit. For a deeper insight into the nature of these additional
transitions, a detailed analysis of spin clusters is performed.
