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(Abridged) We discuss the reliability of neutrino mass constraints, eitheractive or sterile, from the combination of WMAP 9-year or Planck CMB data withBAO measurements from BOSS DR11, galaxy shear measurements from CFHTLenS, SDSSLy-$\alpha$ forest constraints and galaxy cluster mass function from Chandraobservations. To avoid model dependence of the constraints we perform a fulllikelihood analysis for all the datasets employed. As for the cluster dataanalysis we rely on to the most recent calibration of massive neutrino effectsin the halo mass function and we explore the impact of the uncertainty in themass bias and re-calibration of the halo mass function due to baryonic feedbackprocesses on cosmological parameters. We find that none of the low redshiftprobes alone provide evidence for massive neutrinos in combination with CMBmeasurements, while a larger than $2\sigma$ detection of non zero neutrinomass, either active or sterile, is achieved combining cluster or shear datawith CMB and BAO measurements. The preference for massive neutrino is larger inthe sterile neutrino scenario, and for the combination of Planck, BAO, shearand cluster datasets we find that the vanilla $\Lambda$CDM model is rejected atmore than $3\sigma$ and a sterile neutrino mass as motivated by acceleratoranomaly is within the $2\sigma$ errors. Finally, results from the full datacombination reflect the tension between the $\sigma_8$ constraints obtainedfrom cluster and shear data and that inferred from Ly-$\alpha$ forestmeasurements; in the active neutrino scenario for both CMB datasets employed,the full data combination yields only an upper limits on $\sum m_\nu$, whileassuming an extra sterile neutrino we still get preference for non-vanishingmass, $m_s^{\rm eff}=0.26^{+0.22}_{-0.24}$ eV, and dark contribution to theradiation content, $\Delta N_{\rm eff}=0.82\pm0.55$.

Neutrino constraints: what large-scale structure and CMB data are telling us?