Extreme fluctuations in the horizontal geomagnetic field (dB_h/dt) may be generated at the Earth’s surface by electrical currents in the ionosphere and magnetosphere. Using a global database of 125 magnetometers covering several decades we present occurrence statistics for fluctuations exceeding the 99.97^th percentile (P_99.97) for both ramp changes (R_n) and the root-mean-square (S_n) of fluctuations over periods, τ, from 1 to 60 min and describe their variation with geomagnetic latitude and magnetic local time (MLT). Rates of exceedance are explained by reference to the magneto-ionospheric processes dominant in different latitude and MLT sectors, including ULF waves, interplanetary shocks, auroral substorm currents, and travelling convection vortices. By fitting Generalised Pareto tail distributions above P_99.97 we predict return levels (RLs) for R_n and S_n over return periods up to 500 years. P_99.97 and RLs increase monotonically with frequency (1/τ) (with a few exceptions at auroral latitudes) and this is well modelled by quadratic functions whose coefficients vary smoothly with latitude. For UK magnetometers providing 1-s cadence measurements, the analysis is extended to cover periods from 1 to 60 seconds and empirical Magnetotelluric Transfer functions are used to predict percentiles and return levels of the geoelectric field over a wide frequency range (2x10^-4 to 4x10^-2 Hz) assuming a sinusoidal field fluctuation. These results help identify the principal causes of field fluctuations leading to extreme geomagnetically induced currents (GIC) in ground infrastructure over a range of timescales and they inform the choice of frequency dependence to use with dB_h/dt as a GIC proxy.