We investigate nuclear light profiles in 135 ATLAS^(3D) galaxies for which the Hubble Space Telescope (HST) imaging is available and compare them to the large-scale kinematics obtained with the SAURON integral-field spectrograph. Specific angular momentum, λ_R, correlates with the shape of nuclear light profiles, where, as suggested by previous studies, cores are typically found in slow rotators and core-less galaxies are fast rotators. As also shown before, cores are found only in massive galaxies and only in systems with the stellar mass (measured via dynamical models) M ≳ 8 × 10^(10) M⊙. Based on our sample, we, however, see no evidence for a bimodal distribution of nuclear slopes. The best predictor for finding a core is based on the stellar velocity dispersion within an effective radius, σ_e, and specific angular momentum, where cores are found for λ_R ≲ 0.25 and σ_e ≳ 160 km s^(−1). We estimate that only about 10 per cent of nearby early-type galaxies contain cores. Furthermore, we show that there is a genuine population of fast rotators with cores. We also show that core fast rotators are morphologically, kinematically and dynamically different from core slow rotators. The cores of fast rotators, however, could harbour black holes of similar masses to those in core slow rotators, but typically more massive than those found in core-less fast rotators. Cores of both fast and slow rotators are made of old stars and found in galaxies typically lacking molecular or atomic gas (with a few exceptions). Core-less galaxies, and especially core-less fast rotators, are underluminous in the diffuse X-ray emission, but the presence of a core does not imply high X-ray luminosities. Additionally, we postulate (as many of these galaxies lack HST imaging) a possible population of core-less galaxies among slow rotators, which cannot be explained as face-on discs, but comprise a genuine sub-population of slow rotators. These galaxies are typically less massive and flatter than core slow rotators, and show evidence for dynamical cold structures and exponential photometric components. Based on our findings, major non-dissipative (gas-poor) mergers together with black hole binary evolution may not be the only path for formation of cores in early-type galaxies. We discuss possible processes for formation of cores and their subsequent preservation
