  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.
