Context. The interstellar medium (ISM) on all scales is full of structures that can be used as tracers of processes that feed turbulence. Aims. We used H I survey data to derive global properties of the angular power distribution of the local ISM. Methods. HI4PI observations on an nside = 1024 HEALPix grid and Gaussian components representing three phases, the cold, warm, and unstable lukewarm neutral medium (CNM, WNM, and LNM), were used for velocities |vLSR|≤ 25 km s−1. For high latitudes |b| 〉 20° we generated apodized maps. After beam deconvolution we fitted angular power spectra. Results. Power spectra for observed column densities are exceptionally well defined and straight in log-log presentation with 3D power law indices γ ≥−3 for the local gas. For intermediate velocity clouds (IVCs) we derive γ = −2.6 and for high velocity clouds (HVCs) γ = −2.0. Single-phase power distributions for the CNM, LNM, and WNM are highly correlated and shallow with γ ~−2.5 for multipoles l ≤ 100. Excess power from cold filamentary structures is observed at larger multipoles. The steepest single-channel power spectra for the CNM are found at velocities with large CNM and low WNM phase fractions. Conclusions. The phase space distribution in the local ISM is configured by phase transitions and needs to be described with three distinct different phases, being highly correlated but having distributions with different properties. Phase transitions cause locally hierarchical structures in phase space. The CNM is structured on small scales and is restricted in position-velocity space. The LNM as an interface to the WNM envelops the CNM. It extends to larger scales than the CNM and covers a wider range of velocities. Correlations between the phases are self-similar in velocity.