The evolution of cosmic loops (and bubbles) in an inflationary background is governed by a hyperbolic fixed point. Associated with the four distinguished kinematical categories are loops which have or could not have undergone a world-sheet big bang, and which eventually collapse or else keep growing. In particular we discuss spontaneous creation of baby loops, and quantum nucleation of horizon-size loops. World-sheet scalar potentials are invoked to account for the electromagnetic self-interactions of charged and/or superconducting loops. The electromagnetic repulsion at short distances then acts to exclude stationary breathing configurations provided their total charge Q is above a certain critical value set by the string tension/Hubble scale ratio. However, below that critical value there exist charged loops which classically are not inflated by inflation. Quantum-mechanical tunneling renders the classically stable charged loops unstable, and the associated Basu-Guth-Vilenkin amplitude gets a contribution proportional to Q2 log Q2 for small Q.