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In this paper, we extend and strengthen classical estimates for singular values of integral operators originally due to Cwikel, Birman and Solomyak. Suppose that A1 and A2 are two semifinite von Neumann algebras with representations p1 : A1 → B(H) and p2 : A2 → B(H) such that ... Our first result asserts that the inequality... holds in any interpolation space E for (L2, L8). In the special case, when A1 = A2 = L8(Rd) and p1, p2 are given by multipliers and Fourier multipliers, respectively, our result yields a (strengthened version) of well-known Cwikel estimates [Cwikel, Ann. of Math. (2) 106 (1977) 93-100]. We demonstrate further the applicability of our result by considering noncommutative Euclidean space (Moyal plane) and magnetic Laplacian. Our second direction relates to Birman-Solomyak estimates for interpolation space E for the (quasi)-Banach couple (lp, l2), p < 2. In this setting, our technique yields substantial strengthening of results from [Birman and Solomyak, Russian Math. Surveys 32 (1977) 15-89] and Chapter VI of Simon [Trace ideals and their applications (American Mathematical Society, Providence, RI, 2005)]. Finally, we provide Cwikel-Birman-Solomyak estimates for the crucial case of weak Schatten p-ideals, 1 < p < 2, in the setting of noncommutative Euclidean space. [ABSTRACT FROM AUTHOR]