Most applications of nanocarbons, such as carbon nanotubes and graphene, require that they are well-separated and well-dispersed in a liquid phase. Intensive efforts have been put on exfoliating and dispersing nanocarbons in aqueous solvents, typically using amphiphilic dispersants and sonication/centrifugation procedures, alongside a drive to fundamentally understand and rationally optimize these processes. Herein, we employed a robust method to separate and disperse multiwalled carbon nanotubes (MWNTs), and graphene nanoplatelets (GnPs) either from bulk graphite or from pre-formed GnP powders, using rigorously controlled processing conditions. An ionic (sodium cholate) and a nonionic (Triton X-100) surfactant were used as dispersants. Our aim was to determine high-precision dispersibility curves (concentration of dispersed nanomaterial versus initial surfactant concentration) for the different nanocarbon/dispersant systems, characterize morphologically the dispersed particles and compare the mechanisms of exfoliation of 1D and 2D nanocarbons at molecular level. Typically bell-shaped dispersibility curves with a plateau were obtained, and from the latter several quantitative metrics were extracted that permitted reliable comparisons between nanocarbon/surfactant systems. Scanning electron and atomic force microscopies allowed to characterize the suspended particles in the as-obtained dispersions, namely the MWNT bundle width and GnP dimensions (mean lateral size and layer number). Under fixed conditions (in particular, delivered energy per carbon mass), MWNTs are dispersed in much higher yields, by two orders of magnitude, than GnPs. However, and significantly, a master curve for the dispersibility was obtained, implying that common fundamental features underpin the dispersing process, irrespective of nanocarbon (1D or 2D) or surfactant (ionic or nonionic) types.