You can think about some critical phenomena is in terms of analytic continuation and Fisher zeros. As you probably know, the Taylor series expansion an analytic function can only converge within a disk that does not contain singularities. However, you can find Taylor expansions by 'working around' the singularity by means of analytic continuation.
Fisher (and others) realized that the boundary between two phases is separated by a line of zeros. Even if you know a thermodynamic function exactly in one part of the phase diagram, you cannot analytic continuate into another. See Fig. 1.
I mention this because it sounds similar to a paper and talk I recently heard by Anatoli Polkovnikov, who was asking similar questions in regard to a dynamic phase transition.
If that doesn't help, other signs to look for are:
- Critical slowing: it takes longer and longer for dynamics to converge. This applies to simulations as well, which is a good hint!
- A point where correlations are algebraic, not exponential.
- Scaling of dynamical phenomena, such as quenches or ramps. This is another one where simulations might help out.