In this paper we consider the support properties of locally linearly independent refinable function vectors $Phi= (phi_1, ldots, phi_r)^T$. We propose an algorithm for computing the global support of the components of $Phi$. Further, for $Phi=(phi_1, phi_2)^T$ we investigate the supports, especially the possibility of holes of refinable function vectors if local linear independence is assumed. Finally, we give some necessary conditions for local linear independence in terms of rank conditions for special matrices given by the refinement mask. But we are not able to give a final answer to the question whether a locally linearly independent function vector can have more than one hole.