  We report combined magnetic susceptibility, dielectric constant, nuclear
quadruple resonance (NQR) and zero-field nuclear magnetic resonance (NMR)
measurements on single crystals of multiferroics CuBr$_2$. High quality of the
sample is demonstrated by the sharp magnetic and magnetic-driven ferroelectric
transition at $T_N=T_C\approx$ 74~K. The zero-field $^{79}$Br and $^{81}$Br NMR
are resolved below $T_N$. The spin-lattice relaxation rates reveal charge
fluctuations when cooled below 60~K. Evidences of an increase of NMR linewidth,
a reduction of dielectric constant, and an increase of magnetic susceptibility
are also seen at low temperatures. These data suggest an emergent instability
which competes with the spiral magnetic ordering and the ferroelectricity.
Candidate mechanisms are discussed based on the quasi-one-dimensional (1D)
nature of the magnetic system.
