Abstract: Scalar-tensor theories of gravity are extensions of General Relativity including an extra, non-minimally coupled scalar degree of freedom. A wide class of these theories, albeit indistinguishable from GR in the weak field regime, can predict a radically different phenomenology for neutron stars, due to the existence of a non-perturbative, strong-field effect referred to as spontaneous scalarization. This effect is known to occur for theories where the linear effective coupling beta between the scalar and matter fields is sufficiently negative, and has been strongly constrained by pulsar timing observations. In this talk, I discuss the possibility of testing scalar-tensor theories in the highly unconstrained region of coupling functions with beta>0, based on the fact that sufficiently compact neutron stars would be subject to a tachyonic-like instability according to these theories. In the case of beta<0, this instability is understood to be simply the test-field-limit manifestation of spontaneous scalarization; however, in the beta>0 case it can lead to different outcomes. I will discuss the equation of state dependence of this instability, its possible end-states and observational signatures.