In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.