The spin-dependent transport in multiwall carbon nanotubes, obtained by chemical vapour deposition in porous alumina membranes, has been investigated. The zero-bias anomaly is shown to follow the typical power law GT(-alpha) (eV/kT). In the nanotubes contacted with Ni pads, the magnetoresistance due to the spin-polarised current (SD-MR) is destroyed. In the case of those contacted with Co pads, however, the SD-MR is observed. These measurements show that both the observation of a typical scaling law of the conductance (for nanotubes tip to 1.5 mum) and a short spin diffusion length (less than 0.15 mum with nickel contacts and 0.7 mum with cobalt contacts) coexist through the nanotube. This observation may be interpreted in terms of a reminiscence of the Luttinger-liquid behaviour with spin-charge separation.