Quantum vortices in weakly coupled superfluids are characterized by a large healing length, such that a large number of particles reside within the vortex core. Their dynamics are governed by the Gross-Pitaevskii equations deep within the core, which dictate that the superfluid velocity diverges to a singular point at the vortex center. This singularity is topologically protected by the superfluid's vorticity quantization. We analyze the motion of a vortex in a 2D spinless superfluid, and find that the vortex core is in fact unstable to pertubations that obey the topological constraint. In practice, the point singularity deforms into a line singularity, proportional to the applied Magnus force. This core deformation is described by weak solutions of the Gross-Pitaevskii equations. The core deformation produces an anomalously large dipole moment associated with an anomalously large mass. As a result, the vortex motion exhibits narrow resonances lying within the phonon part of the spectrum. The deformation also impacts the dynamics of few-vortex systems, leading to chaotic behaviour of vortex-antivortex dipoles.