We find stable singularity-free cosmological solutions in non-flat
Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime in the context of
Ho\v{r}ava-Lifshitz (HL) theory. Although we encounter the negative squared
effective masses of the scalar perturbations in the original HL theory, the
behaviors can be remedied by relaxing the projectability condition. In our
analysis, the effects from the background dynamics are taken into account as
well as the sign of the coefficients in the quadratic action for perturbations.
More specifically, we give further classification of the gradient
stability/instability into five types. These types are defined in terms of the
effective squared masses of perturbations $\mathcal{M}^2$, the effective
friction coefficients in perturbation equations $\mathcal{H}$ and these
magnitude relations $|\mathcal{M}^2|/\mathcal{H}^2$. Furthermore, we indicate
that oscillating solutions possibly show a kind of resonance especially in open
FLRW spacetime. We find that the higher order spatial curvature terms with
Lifshitz scaling $z=3$ are significant to suppress the instabilities due to the
background dynamics.