  We extend previous theoretical studies of the contribution of fluctuating
Cooper pairs to the persistent current in superconducting rings subjected to a
magnetic field. For sufficiently small rings, in which the coherence length
$\xi$ exceeds the radius $R$, mean field theory predicts the emergence of a
flux-tuned quantum critical point separating metallic and superconducting
phases near half-integer flux through the ring. For larger rings with $R\gtrsim
\xi$, the transition temperature is periodically reduced, but superconductivity
prevails at very low temperatures. We calculate the fluctuation persistent
current in different regions of the metallic phase for both types of rings.
Particular attention is devoted to the interplay of the angular momentum modes
of the fluctuating order parameter field. We discuss the possibility of using a
combination of different pair-breaking mechanisms to simplify the observation
of the flux-tuned transition in rings with $\xi>R$.
