  We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose
weight has a one-cut regular potential and Fisher-Hartwig singularities. We
restrict our attention to the case where the associated equilibrium measure
possesses either one soft edge and one hard edge (Laguerre-type) or two hard
edges (Jacobi-type). We also present some applications in the theory of random
matrices. In particular, we can deduce from our results asymptotics for
partition functions with singularities, central limit theorems, correlations of
the characteristic polynomials, and gap probabilities for (piecewise constant)
thinned Laguerre and Jacobi-type ensembles. Finally, we mention some links with
the topics of rigidity and Gaussian multiplicative chaos.
