It seems to have disappeared. The menu Edit/Search doesn't work, either.
I've filed a launchpad bug-report.
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Gary Lasker from Canoical contacted me through launchpad. He asked me to run Ubuntu Software Center from a terminal using the command line.
I did that and the search box is back. There were no updates in the interim so it possibly was a one time thing (so far).
Thanks to Gary for getting back to me quickly!