This paper presents a method of diagnosing variance components of process error sources in singular manufacturing systems. The singularity problem is studied and the cause examined in the context of fixture error diagnosis in multi-station assembly processes. The singularity problem results in nondiagnosable fixture errors when standard least-squares (LS) estimation methods are used. This paper suggests a reformulation of the original error propagation model into a covariance relation. The LS criterion is then applied directly to the sample covariance matrix to estimate the variance components. Diagnosability conditions for this variance LS estimator are derived, and it is demonstrated that certain singular systems that are not diagnosable using traditional LS methods become diagnosable with the variance LS estimator. Modified versions that improve the accuracy of the variance LS estimator are also presented. The various procedures are thoroughly contrasted, in terms of accuracy and diagnosability. The results are illustrated with examples from panel assembly, although the application of the approach and the conclusions extend to more general discrete-part manufacturing processes where fixtures are used to ensure dimensional accuracy of the final product.