Abstract: A key goal in the fields of quantum optomechanics and electromechanics is to measure and hopefully control truly quantum behaviour in a "large" mechanical resonator. In this talk, I will start by giving a quick introduction to these two rapidly growing fields. I will then discuss recent experimentally-motivated theoretical work which calculates the full statistics of low-frequency energy fluctuations of a driven, quantum resonator. Surprisingly, these fluctuations are highly non-classical: they are most naturally described by a quasi-probability distribution, which can be negative. Somewhat surprisingly, this effect has a close relationship to the statistics of electronic charge transfer (so-called "full counting statistics") in mesoscopic superconducting conductors. I will discuss how these effects might be measured, and how they represent a kind of non-classical behaviour similar to the violation of a Leggett-Garg inequality. While the emphasis is on phonons, our results apply equally as well to the photon fluctuations of a driven cavity.