  We investigate two dark energy cosmological models (i.e., the $\Lambda$CDM
and $\phi$CDM models) with massive neutrinos assuming two different neutrino
mass hierarchies in both the spatially flat and non-flat scenarios, where in
the $\phi$CDM model the scalar field possesses an inverse power-law potential,
$V(\phi)\propto {\phi}^{-\alpha}$ ($\alpha>0$). Cosmic microwave background
data from Planck 2015, baryon acoustic oscillations data from 6dFGS, SDSS-MGS,
BOSS-LOWZ and BOSS CMASS-DR11, the JLA compilation of Type Ia supernova
apparent magnitude observations, and the Hubble Space Telescope $H_0$ prior,
are jointly employed to constrain the model parameters. We first determine
constraints assuming three species of degenerate massive neutrinos. In the
spatially flat (non-flat) $\Lambda$CDM model, the sum of neutrino masses is
bounded as $\Sigma m_{\nu} < 0.165 (0.299)$ eV at 95% confidence level (CL).
Correspondingly, in the flat (non-flat) $\phi$CDM model, we find $\Sigma
m_{\nu} < 0.164 (0.301)$ eV at 95% CL. The inclusion of spatial curvature as a
free parameter results in a significant broadening of confidence regions for
$\Sigma m_{\nu}$ and other parameters. In the scenario where the total neutrino
mass is dominated by the heaviest neutrino mass eigenstate, we can obtain the
similar conclusions as those obtained in the degenerate neutrino mass scenario.
In addition, the results show that the bounds on $\Sigma m_{\nu}$ based on two
different neutrino mass hierarchies have insignificant differences in the
spatially flat case for both the $\Lambda$CDM and $\phi$CDM models, however,
the corresponding differences are larger in the non-flat case.
