We present a detailed theoretical investigation on the magnetic properties of small single-layered Fe, Co, and Ni clusters deposited on Ir(111), Pt(111), and Au(111). For this, a fully relativistic ab initio scheme based on density functional theory has been used. We analyze the element, size, and geometry specific variations of the atomic magnetic moments and their mutual exchange interactions as well as the magnetic anisotropy energy in these systems. Our results show that the atomic spin magnetic moments in the Fe and Co clusters decrease almost linearly with increasing coordination number on all three substrates, while the corresponding orbital magnetic moments appear to be much more sensitive to the local atomic environment. The isotropic exchange interaction among the cluster atoms is always very strong for Fe and Co exceeding the values for bulk bcc Fe and hcp Co, whereas the anisotropic Dzyaloshinski-Moriya interaction is, in general, one or two orders of magnitude smaller when compared to the isotropic one. For the magnetic properties of Ni clusters, the magnetic properties can show quite a different behavior, and we find in this case a strong tendency towards noncollinear magnetism.