Neural operators have emerged as a promising paradigm for learning solution operators of partial differential equations (PDEs) directly from data. Existing methods, such as those based on Fourier or graph techniques, make strong assumptions about the structure of the kernel integral operator, assumptions which may limit expressivity. We present SVD-NO, a neural operator that explicitly parameterizes the kernel by its singular-value decomposition (SVD) and then carries out the integral directly in the low-rank basis. Two lightweight networks learn the left and right singular functions, a diagonal parameter matrix learns the singular values, and a Gram-matrix regularizer enforces orthonormality. As SVD-NO approximates the full kernel, it obtains a high degree of expressivity. Furthermore, due to its low-rank structure the computational complexity of applying the operator remains reasonable, leading to a practical system. In extensive evaluations on five diverse benchmark equations, SVD-NO achieves a new state of the art. In particular, SVD-NO provides greater performance gains on PDEs whose solutions are highly spatially variable.