The behaviour of positive muons ( mu ⁺) has been studied in Cu, V, Nb and Ta single crystals, using the muon spin rotation ( mu SR) technique. The measurements were performed in the temperature regime 5-365K, in transverse external magnetic fields of 50-5500 G. In FCC Cu the mu ⁺ diffusional motion can be described in terms of quantum mechanical jump processes of a small polaron among octahedral interstitial sites. An orientation average leads to an activation energy of E=79.2+or-3.1 meV, a pre-exponential factor of D(0)=(1.66+or-0.33)*10^-8 cm² s^-1 and a tunnelling matrix element of J=(18.3+or-1.8)*10^-6 eV. The mu ⁺ total volume expansion was found to be Delta V=3.91 AA³. Furthermore, the electric field gradient (EFG) at the mu ⁺ nearest-neighbour nuclei has been determined as q=0.30 AA^-3. In BCC V, Nb and Ta the mu ⁺ diffusion is trap-limited. As a function of the temperature the mu ⁺ probe different classes of trapping sites associated with host metal impurities. At low temperature, before decaying, the mu ⁺ always end up in a trapped state which shows that the intrinsic mu ⁺ motion between traps is fast and seems to reflect properties of a band diffusion.