We consider the high radio-frequency (15-353GHz) properties and variability of 35 brightest cluster galaxies (BCGs). These are the most core-dominated sources drawn from a parent sample of more than 700 X-ray selected clusters, thus allowing us to relate our results to the general population. We find that >=6.0 percent of our parent sample (>=15.1 percent if only cool-core clusters are considered) contain a radio source at 150GHz of at least 3mJy (~1x10^23^W/Hz at our median redshift of z~0.13). Furthermore, >=3.4 percent of the BCGs in our parent sample contain a peaked component (Gigahertz Peaked Spectrum, GPS) in their spectra that peaks above 2GHz, increasing to >=8.5 percent if only cool-core clusters are considered. We see little evidence for strong variability at 15GHz on short (week-month) time-scales although we see variations greater than 20 percent at 150GHz over six-month time frames for 4 of the 23 sources with multi-epoch observations. Much more prevalent is long-term (year-decade time-scale) variability, with average annual amplitude variations greater than 1 percent at 15GHz being commonplace. There is a weak trend towards higher variability as the peak of the GPS-like component occurs at higher frequency. We demonstrate the complexity that is seen in the radio spectra of BCGs and discuss the potentially significant implications of these high-peaking components for Sunyaev-Zel'dovich cluster searches.