Measurements of the differential and double-differential Drell-Yan cross sections are presented using an integrated luminosity of 4.5 (4.8) inverse femtobarns in the dimuon (dielectron) channel of proton-proton collision data recorded with the CMS detector at the LHC at sqrt(s) = 7 TeV. The measured inclusive cross section in the Z-peak region (60-120 GeV) is sigma(ell ell) = 986.4 +/- 0.6 (stat.) +/- 5.9 (exp. syst.) +/- 21.7 (th. syst.) +/- 21.7 (lum.) pb for the combination of the dimuon and dielectron channels. Differential cross sections d[sigma]/d[m] for the dimuon, dielectron, and combined channels are measured in the mass range 15 to 1500 GeV and corrected to the full phase space. Results are also presented for the measurement of the double-differential cross section d2[sigma]/d[m]d[abs(y)] in the dimuon channel over the mass range 20 to 1500 GeV and absolute dimuon rapidity from 0 to 2.4. These measurements are compared to the predictions of perturbative QCD calculations at next-to-leading and next-to-next-to-leading orders using various sets of parton distribution functions.