  Given a compact polarized manifold $(X,L)$, we introduce two new stability
thresholds in terms of singularity types of global quasi-plurisubharmonic
functions on $X$. We prove that in the Fano setting, the new invariants can
effectively detect the K-stability of $X$. We study some functionals of
geodesic rays in the space of K\"ahler potentials by means of the corresponding
test curves. In particular, we introduce a new entropy functional of
quasi-plurisubharmonic functions and relate the radial entropy functional to
this new entropy functional.
