We describe interacting mixtures of ultracold bosonic and fermionic atoms in harmonically confined optical lattices. For a suitable choice of parameters we study the emergence of superfluid and Fermi-liquid (noninsulating) regions out of Bose-Mott and Fermi-band insulators due to finite boson and fermion hoppings. We obtain the shell structure for the system and show that angular momentum can be transferred to the noninsulating regions from Laguerre-Gaussian beams, which combined with Bragg spectroscopy can reveal all superfluid and Fermi-liquid shells.