The majority of pulsar glitches are detected via radio observations, but then again the majority of pulsar observations are conducted in the radio so this is not so surprising. X-ray and gamma-ray pulsars are certainly included and are an important minority - for example, one of the most prolifically glitching pulsars, J0537-6910 (45 glitches in the Jodrell Bank catalogue at time of writing!), is timed using X-ray observations. I'm not aware of any glitch discoveries being made with optical data, but I don't see any reason in principle that they couldn't contribute, I expect it's more a matter of a relative lack of optical pulsars and active optical timing programmes.
They do not have to be caught "in the act", and this is fortunate! Glitches occur infrequently and on very short timescales, and observations of almost all pulsars are not dense enough to have a good chance of catching a glitch during active observation*. Glitches are usually noticed after the fact (sometimes long after the fact) by inspecting the timing residuals of anywhere from many days to many years of data surrounding the glitch and noticing some feature which indicates a sharp change in the frequency.
* A very interesting exception to this was reported a few years ago wherein observations at Mt. Pleasant in Hobart, Australia caught the Vela pulsar glitching while recording every individual pulse (as opposed to recording data folded at the pulse period, which is more typical for routine pulsar observations). This is described in this paper (apologies for the paywall!).