Origami-inspired design has recently emerged as a major thrust area of research in the fields of science and engineering. One such design utilizes Kresling-pattern origami to construct nonlinear springs that can act as mechanical bit memory switches, wave guides, fluidic muscles, and vibration isolators. The main objective of this work is to characterize the static equilibria of such springs, their stability, and bifurcations as the geometric parameters of the Kresling pattern are varied. To this end, a mathematical model which assumes that the different panels can be represented by axially deformable truss elements is adopted. The adopted model demonstrates that the shape of the potential energy of the spring is very sensitive to changes in its geometric parameters. This causes the static configuration to undergo several bifurcations as one or more of the geometrical parameters are varied. In particular, it is shown that the geometric parameter space of the Kresling pattern can be divided into five regions, each of which results in a qualitatively different spring behavior. Results of the axial truss model are verified experimentally demonstrating that, for the most part, the model is capable of predicting the loci and bifurcations of the spring's equilibria. Nevertheless, it is also observed that, away from the equilibrium points, the quasistatic behavior of the spring is not well-approximated by the axial truss model. To overcome this issue, a modified model is developed which accounts for (i) the rotary stiffness of the creases, (ii) self avoidance due to panel contact at small angles between the panels, and (iii) buckling of the creases under compressive loads. It is shown that the modified model is capable of providing a better overall qualitative approximation of the quasistatic behavior.