Thanks Audra.
Unfortunately I do not know when the search problem started - it was only when a user alerted me to it that I realised there was a problem. But the absence of earlier reports makes me think that it is within the last month - during which I have:
1. Upgraded from Acajoom Free to Plus
2. Added JoomlaCaptcha
3. Added ProjectFork
These are all different to my other J sites.
I thought the captcha may be the culprit because it seems less mainstream than other extensions, involved patching some of the core and may have intercepted an entered search term before it was recognised by the search function. The search statistics are showing a growing number of searches but all with an empty search term field. But disabling the captcha, and now uninstalling it completely, has not fixed the problem. Upgrading to 1.5.11 might overwrite any core patching that the captcha uninstall did not do - though it reported a clean uninstall.
Neither Acajoom or ProjectFork seem to be likely culprits, but testing their removal and the 1.5.11 upgrade are on my list for next week on a new test site.
In terms of other oddities on this site, I did have a hiccup with JoomlaPack falling over - which I tracked back to a huge cache folder (some 25000+ files) belonging to JomComment - which for some reason was not being automatically cleared. I have captcha turned of for this Comment component and possibly related I noticed that there were some 28900 records in the jos_captcha_session table. The site has received a number of bot registrations and comments - hence the captchas. I am clearing the JomComment cache files.
I have just re-enabled the other search plugins as you suggested, but they were all on when the problem was identified.
Yes the site can go slow every now and then - it is on a shared server on a cheaper hosting service - hope Brad doesn't read this. One of my to do tasks after Tuesday's meeting is to investigate the Google page speed Firebug addon, but I think the occasional slow speed is generally a server temporary issue.
Thanks for your help - David.