Ever since "Google Instant" was released by Google, the search box occasionally just disappears in Firefox:
This seems to happen right around the moment when Google Instant directs Firefox to a new URL and I happen to be pressing the back button on my Logitech G500 Mouse.
Which is, by the way, a very nice mouse for gaming and general-purpose use for us right-handed folks. Left-handed folks are, unfortunately, so neglected by computer equipment manufacturers.
Anyway, I'm not sure why this bug happens.
There is also another annoying bug with Google Instant that happens when I'm typing quickly. It switches from the main page to the Google Instant page. The search box is there with my search but complete search results refuse to show up and clicking the search button does nothing. Starting over and running the exact same search causes search results to show up.
Google is one of those companies that doesn't publish their e-mail addresses anywhere. I do have a few direct e-mail contacts but none of them are exactly appropriate for reporting bugs with their core product (search).