Cosmological neutrino mass and abundance measurements are reaching unprecedented precision. Testing their stability versus redshift and scale is a crucial issue, as it can serve as a guide for optimizing ongoing and future searches. Here, we perform such analyses, considering a number of redshift, scale, and redshift-and-scale nodes. Concerning the $k$-space analysis of $\sum m_{\nu }$, cosmic microwave background (CMB) observations are crucial, as they lead the neutrino mass constraints. Interestingly, some data combinations suggest a nonzero value for the neutrino mass with $2\sigma$ significance. The most constraining bound we find is $\sum m_{\nu }<0.54\mathrm{eV}$ at 95% confidence level (CL) in the $[{10}^{-3},{10}^{-2}]h/\mathrm{Mpc}$ $k$-bin, a limit that barely depends on the data combination. Regarding the redshift- and scale-dependent neutrino mass constraints, high redshifts ($z>100$) and scales in the range $[{10}^{-3},{10}^{-1}]h/\mathrm{Mpc}$ provide the best constraints. The least constraining bounds are obtained at very low redshifts [0, 0.5] and also at very small scales ($k>0.1h/\mathrm{Mpc}$) due to the absence of observations. Highly relevant is the case of the [100, 1100], $[{10}^{-2},{10}^{-1}]h/\mathrm{Mpc}$ redshift-scale bin, where a $2-3\sigma$ evidence for a nonzero neutrino mass is obtained for all data combinations. The bound from CMB alone at 68% CL is ${0.63}_{-0.24}^{+0.20}\mathrm{eV}$, and the one for the full dataset is ${0.56}_{-0.23}^{+0.20}\mathrm{eV}$, clearly suggesting a nonzero neutrino mass at these scales, possibly related to a deviation of the integrated Sachs-Wolfe amplitude in this redshift range. Concerning the analysis of $N_{\mathrm{eff}}$ in the $k$-space, at intermediate scales ranging from $k={10}^{-3}h/\mathrm{Mpc}$ to $k={10}^{-1}h/\mathrm{Mpc}$, accurate CMB data provide very strong bounds, the most robust one being $N_{\mathrm{eff}}=3.09\pm 0.14$, comparable to the standard expected value without a $k$-bin analysis. If a nonzero neutrino mass is considered, the bounds on the $N_{\mathrm{eff}}$ values at the different $k$-bins are largely unaffected, and the 95% CL tightest limit we find for the neutrino mass in this case is $\sum m_{\nu }<0.205\mathrm{eV}$ from the full dataset. Finally, the $z$ and $k$ analyses of $N_{\mathrm{eff}}$ indicate a high constraining power of cosmological observations at high redshifts and intermediate scales $[{10}^{-2},{10}^{-1}]h/\mathrm{Mpc}$ when extracting the binned values of this parameter.

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• Baryon acoustic oscillations
• Cosmic microwave background
• Cosmological parameters
• Evolution of the Universe
• Neutrino mass