You supposedly want to identify which from a set of candidate distributions fits the observed data best. Generally, goodness-of-fit tests have an odd logic and should be avoided altogether. For example, Zuur et.al. (2010) recommend visual checks to check distributional assumptions.

An alternative is formal model comparison using the Akaike Information Criterion (AIC). For this purpose you fit the data to the candidate distributions using maximum likelihood. The smaller AIC indicates the better fit, taking parsimony into account. If you are an R user, the VGAM package offers almost any statistical ddistribution you can think of.

In any case, make sure to use the same software package for the maximum likelihood estimation, otherwise the AIC values may not be comparable due to some constants being included or excluded.

Zuur, A.F., Ieno, E.N., and Elphick, C.S. A protocol for data exploration to avoid

common statistical problems. Methods in Ecology and Evolution 1, 1 (2010), 3–14.