I will present a desingularization technique for symplectic (and contact) structures with singularities which appear modelling some problems in Celestial Mechanics and describe several applications to the study of their Hamiltonian (and Reeb) Dynamics. In these motivating examples the singularities are associated to the line at infinity or collision set and are an outcome of regularization techniques. These singular symplectic structures can be formalized as smooth Poisson structures which are symplectic away from a hypersurface and satisfy some transversality properties. The desingularization technique or "deblogging" (joint work with Victor Guillemin and Jonathan Weitsman) associates a family of symplectic structures to singular symplectic structures with even exponent (the so-called $b^{2k}$-symplectic structures) and a family of folded symplectic structures for odd exponent ($b^{2k+1}$-symplectic structures) and has good convergence properties. This procedure generalizes to its odd-dimensional counterpart (joint work with Cédric Oms) and puts in the same picture different geometries: symplectic, folded-symplectic, contact and Poisson geometry. The applications of this "desingularization kit" include the construction of action-angle coordinates for integrable systems, the study of their perturbation (KAM theory) and the existence of periodic orbits. Time permitting, we will end up this talk with a proof of a singular Weinstein conjecture for singular contact manifolds and with some open questions in Hamiltonian and Reeb Dynamics related to the motivating problems in Celestial Mechanics.