Emulsion and bulk freezing experiments were performed to investigate immersion ice nucleation on clay minerals in pure water, using various kaolinites, montmorillonites, illites as well as natural dust from the Hoggar Mountains in the Saharan region. Differential scanning calorimeter measurements were performed on three different kaolinites (KGa-1b, KGa-2 and K-SA), two illites (Illite NX and Illite SE) and four natural and acid-treated montmorillonites (SWy-2, STx-1b, KSF and K-10). The emulsion experiments provide information on the average freezing behaviour characterized by the average nucleation sites. These experiments revealed one to sometimes two distinct heterogeneous freezing peaks, which suggest the presence of a low number of qualitatively distinct average nucleation site classes. We refer to the peak at the lowest temperature as "standard peak" and to the one occurring in only some clay mineral types at higher temperatures as "special peak". Conversely, freezing in bulk samples is not initiated by the average nucleation sites, but by a very low number of "best sites". The kaolinites and montmorillonites showed quite narrow standard peaks with onset temperatures 238 K < T^std_on < 242 K and best sites with averaged median freezing temperature T^med_best = 257 K, but only some featuring a special peak (i.e. KSF, K-10, K-SA and SWy-2) with freezing onsets in the range 240-248 K. The illites showed broad standard peaks with freezing onsets at 244 K < T^std_on < 246 K and best sites with averaged median freezing temperature T^med_best = 262 K. The large difference between freezing temperatures of standard and best sites shows that characterizing ice nucleation efficiencies of dust particles on the basis of freezing onset temperatures from bulk experiments, as has been done in some atmospheric studies, is not appropriate. Our investigations demonstrate that immersion freezing temperatures of clay minerals strongly depend on the amount of clay mineral present per droplet and on the exact type (location of collection and pre-treatment) of the clay mineral. We suggest that apparently contradictory results obtained by different groups with different setups are indeed in good agreement when only clay minerals of the same type and amount per droplet are compared. The natural sample from the Hoggar Mountains, a region whose dusts have been shown to be composed mainly of illite, showed very similar freezing characteristics (standard and best) to the illites. Relating the concentration of best IN to the dust concentration in the atmosphere suggested that the best IN in the Hoggar sample would be common enough downwind of their source region to account for ambient IN number densities in the temperature range of 250-260 K at least during dust events.