  We consider the high radio frequency (15 GHz - 353 GHz) 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% of our parent sample (>15.1% if only cool-core
clusters are considered) contain a radio-source at 150 GHz of at least 3mJy
(~1x10^23 W/Hz at our median redshift of z~0.13). Furthermore, >3.4% of the
BCGs in our parent sample contain a peaked component (Gigahertz Peaked
Spectrum, GPS) in their spectra that peaks above 2 GHz, increasing to >8.5% if
only cool-core clusters are considered. We see little evidence for strong
variability at 15 GHz on short (week-month) time-scales although we see
variations greater than 20% at 150 GHz over 6-month times-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% at 15 GHz 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.
