This purely theoretical work investigates the problem of artificial singularities in camera self-calibration. Self-calibration allows one to upgrade a projective reconstruction to metric and has a concise and well-understood formulation based on the Dual Absolute Quadric (DAQ), a rank-3 quadric envelope satisfying (nonlinear) 'spectral constraints': it must be positive of rank 3. The practical scenario we consider is the one of square pixels, known principal point and varying unknown focal length, for which generic Critical Motion Sequences (CMS) have been thoroughly derived. The standard linear self-calibration algorithm uses the DAQ paradigm but ignores the spectral constraints. It thus has artificial CMSs, which have barely been studied so far. We propose an algebraic model of singularities based on the confocal quadric theory. It allows to easily derive all types of CMSs. We first review the already known generic CMSs, for which any self-calibration algorithm fails. We then describe all CMSs for the standard linear self-calibration algorithm; among those are artificial CMSs caused by the above spectral constraints being neglected. We then show how to detect CMSs. If this is the case it is actually possible to uniquely identify the correct self-calibration solution, based on a notion of signature of quadrics. The main conclusion of this paper is that a posteriori enforcing the spectral constraints in linear self-calibration is discriminant enough to resolve all artificial CMSs.