The frequency range over which a mount can isolate a mass from a vibrating base (or vice versa) is often limited by the mount stiffness required to support the weight of the mass. This compromise can be made more favourable by employing non-linear mounts with a softening spring characteristic such that small excursions about the static equilibrium position result in small dynamic spring forces and a correspondingly low natural frequency. This paper concerns the force-displacement characteristic of a so-called quasi-zero-stiffness (QZS) mechanism which is characterised by an appreciable static stiffness but very small (theoretically zero) dynamic stiffness. The mechanism studied comprises a vertical spring acting in parallel with two further springs which, when inclined at an appropriate angle to the vertical, produce a cancelling negative stiffness effect. Analysis of the system shows that a QZS characteristic can be obtained if the system’s parameters (angle of inclination and ratio of spring stiffnesses) are opportunely chosen. By introducing the additional criterion that the displacement of the system be largest without exceeding a desired (low) value of stiffness an optimal set of parameter values is derived. Under sufficiently large displacements the stiffness of the QZS mechanism can eventually exceed that of the simple mass-spring system and criteria for this detrimental scenario to arise are presented.