Local gauge symmetries cannot break spontaneously, according to Elitzur's theorem, but this leaves open the possibility of breaking some global subgroup of the local gauge symmetry, which is typically the gauge symmetry remaining after certain (e.g. Coulomb or Landau) gauge choices. We show that in an SU(2) gauge-Higgs system such symmetries do indeed break spontaneously, but the location of the breaking in the phase diagram depends on the choice of global subgroup. The implication is that there is no unique broken gauge symmetry, but rather many symmetries which break in different places. The problem is to decide which, if any, of these gauge symmetry breakings is associated with a transition between physically different, confining and non-confining phases. Several proposals − Kugo-Ojima, Coulomb, and monopole condensate − are discussed.