  Spin-glass magnetism confined to individual weakly interacting vortices is
detected in two different families of high-transition-temperature (T_c)
superconductors, but only in samples on the low-doping side of the
low-temperature normal state metal-to-insulator crossover (MIC). Our findings
unravel the mystery of the MIC, but more importantly identify the true location
of the field-induced quantum phase transition (QPT) in the superconducting
state. The non-uniform appearance of magnetism in the vortex state favours a
surprisingly exotic phase diagram, in which spatially inhomogeneous competing
order is stabilized at the QPT, and an `avoided' quantum critical point (QCP)
is realized at zero magnetic field.
