Necessary and sufficient conditions and also simple sufficient conditions are given for the self-adjoint operators associated with the second-order linear differential expression τ(y) 1/w(-(py')'+qy) on [a,b) to have discrete spectrum. Here the coefficients of τ are non-negative and satisfy minimal smoothness conditions. These results follow from compact embedding theorems from a weighted one-dimensional Sobolev space with norm∫ab(p∣f′∣r+q∣f′∣r))1/r into a weighted Banach space with norm(∫abw∣f′∣s)1/s .