We analyze shape phase transitions in two-dimensional algebraic models. We apply our analysis to linear-to-bent transitions in molecules and point out what observables are particularly sensitive to the transition (order parameters). We study numerically the scaling behavior of observables and confirm the dependence of the energy gap for phase transitions of U (n) -SO (n+1) type. We calculate energies of excited states and show their unusual behavior for some values of the Hamiltonian control parameter. This behavior is due to the double-humped nature of the potential and can be associated with the concept of monodromy. Finally, we compute numerically thermodynamic quantities, in particular heat capacities, and show their large variation at and around the critical value of the control parameter. © 2008 The American Physical Society.