Download ccleaner:
http://www.piriform.com/ccleaner

It's free. Use it after every browsing session.
It removes everything....whether you know something is turned on or not.
It will also straighten up the registry, edit start menu programs, and keep the machine running smooth and fast.

it leaves the same traces that any other browser does...  

I do not knowingly take unnecessary risks...  there are enough built in risks...   I normally use my own computer but have an old lap top too.  Everything is set to delete history on closing & I ALWAYS shut it off...  my theory is that it helps fight tracking cookies & spam.   It is extreemly rare but not impossible that some other might use my computer however.  
I hadn't even used the laptop for over a month but had occasion to today.  So I fired it up and after the auto-updates stopped running, I was doing my normal day to day web business.  I clicked on one site & needed to enter my user-name & password...  up popped a list of user names...  including this one.  OMG!!  

It took a bit of head scratching but it was the auto-complete "feature"...  I don't know how it go turned on, but I shut it off.  In IE8  tools> internet options> content tab> auto complete settings

One can never be too careful.
HV

I understand the value of what you're talking about but, so long as you shut the computer off when you're done with it,  and you have a log-on code known only to you so that the desk-top and web browsing functions are available to no-one else, why is that not sufficient protection?  That's what I do and I'd really like to know if there's a flaw in my logic.

Crumbs / trails still need to be cleaned, unless one is running in privacy mode... which again most other browsers have.

as usual, apple users resort to completely irrelevant bashing and insults on a topic that has nothing to do with it.

You wouldn't understand it anyway. It is all in the design. Firefox, Safari, Chrome are all designed different from IE and Windows.

refers to...  Googling it brings up all sorts of Real Estate Ads...  no one has the P/W & I use the In Private setting which is why this was upsetting.   I shared because this was an unanticipated problem...  caught me by surprize.  
I thank everyone for the thoughtful advise.  There is no room for careless or sloppy in my hobbying...  little slip ups are bound to happen but I'm sure going to fight them & minimize the ones I can.   I was alone when I found it...  I stuck with it until I had solved it.
Little mistakes plant the seeds of doubt which can build to suspicion...  then all bets are off!  I repeat, One can never be too careful.
HV

I have my web browser set to clear all history every time I close my internet down... yea, I have to type in the full web addresses EVERY time I get online, but there is never any browsing history.