Google Instant is a new real-time search service from Google, which for some reason does not seem to work in Opera even though it supports all the other major browsers.
But on closer inspection, it turns out that it does work in Opera! All you need to do is to work around Google's browser sniffing and spoof as a different browser, as demonstrated by Bruce Lawson. …
(Watch on YouTube)
So it seems that Google has chosen to exclude Opera (or only include specific browsers). This is rather disappointing, but at least it's possible to work around it.
Someone created a thread in the Google support forum about the issue in case someone wants to voice their opinion on it.