In this article, we derive statistical tools to analyze and distinguish the patterns of genetic variability produced by classical and recent population genetic models related to seed banks. In particular, we are concerned with models described by the Kingman coalescent (K), models exhibiting so-called weak seed banks described by a time-changed Kingman coalescent (W), models with so-called strong seed bank described by the seed bank coalescent (S) and the classical two-island model by Wright, described by the structured coalescent (TI). As the presence of a (strong) seed bank should stratify a population, we expect it to produce a signal roughly comparable to the presence of population structure. We begin with a brief analysis of Wright's $F_{ST}$, which is a classical but crude measure for population structure, followed by a derivation of the expected site frequency spectrum (SFS) in the infinite sites model based on 'phase-type distribution calculus' as recently discussed by Hobolth et al. (2019). Both the $F_{ST}$ and the SFS can be readily computed under various population models, they discard statistical signal. Hence we also derive exact likelihoods for the full sampling probabilities, which can be achieved via recursions and a Monte Carlo scheme both in the infinite alleles and the infinite sites model. We employ a pseudo-marginal Metropolis-Hastings algorithm of Andrieu and Roberts (2009) to provide a method for simultaneous model selection and parameter inference under the so-called infinitely-many sites model, which is the most relevant in real applications. It turns out that this full likelihood method can reliably distinguish among the model classes (K, W), (S) and (TI) on the basis of simulated data even from moderate sample sizes. It is also possible to infer mutation rates, and in particular determine whether mutation is taking place in the (strong) seed bank.