When choosing a statistical test, you generally want to go for one of the more well-known ones, like the chi-square goodness of fit test.That's because more people are going to be able to understand your results, and you have the backing of a slew of researchers before you who have proved that the tests are valid within certain parameters.
That said, there are those rare times when data doesn't neatly fit one of the more popular tests for model fitting, violates one or more assumptions (like the assumption of independence), or is simply too sparse to neatly fit to any common model.
When you run an obscure test, you may--in the worst case scenario--only have a handful of researchers before you who have attempted to study data using a particular test. Therefore, you may not be able to trust your results. If you do choose to use one of the more obscure tests, your rationale for using it should be justified and referenced.
This one picture shows a handful of some relatively obscure tests for model fitting.
D.A. Burn and T.A. Ryan, Jr. (1983). “A Diagnostic Test for Lack of Fit in Regression Models,” ASA 1983 Proceedings of the Statistical Computing Section, pp.286–290.
Christensen, R. Testing Lack of Fit
Mackenzie & Bailey. Assessing the Fit of Site-Occupancy Models
Utts, J. The Rainbow Test for Lack of Fit in Regression. Communication in Statistics- Theory and Methods 11(24):2801-2815
Gustafson, L. Bringing consistency to simulation of population models – Poisson Simulation as a bridge between micro and macro simulation. Mathematical Biosciences.. Volume 209, Issue 2. October 2007, Pages 361-385