http://www.google.com/support/bin/answer.py?answer=465
How do I delete the drop-down list of my past searches?
You can clear your Google search history from the Google search box, the Google Toolbar, or your browser. We've included instructions for all three methods below.
CLEARING YOUR GOOGLE SEARCH BOX HISTORY
The search history that displays in the search box on the Google homepage is stored by your browser, not by Google. Instructions appear below for disabling this search history in common browsers. If your browser isn't listed, we suggest consulting its online support center.
Microsoft Internet Explorer
1. Go to the "Tools" menu.
2. Select "Internet Options" and the "Content" tab.
3. Within the "Personal information" area, select "AutoComplete."
4. Click on "Clear Forms." You can also un-check the "Forms" box in this window to keep this information from being stored in the future.
5. Click "OK" to exit.
Alternatively, you can delete individual entries from your search history by using the Down arrow key to highlight a previous search, then pressing the Delete key once.
MSN
If you're using the Microsoft Network (MSN), changes made in Internet Explorer (IE) will take effect in your MSN browser. After making changes in IE, you'll need to sign out of and back into MSN for the changes to take effect.
Mozilla Firefox
1. Go to the "Tools" menu.
2. Select "Options" > "Privacy" > "Saved Form Information."
3. Click "Clear" and un-check the box.
4. Click "OK" to exit.
Safari
1. Go to the "Safari" menu and select "Preferences."
2. From the menus that pop up, select "Autofill."
3. Go to "Other forms" line and select the "Edit" button.
4. From the list of sites for which AutoFill is currently being applied, select "Google.com" > "Remove" > "Done."
CLEARING YOUR GOOGLE TOOLBAR HISTORY
To clear your search history from the drop-down text box on the Google Toolbar:
1. Click on the Google logo on the Toolbar to access the drop-down menu.
2. Select the "Clear Search History" option.
If you'd like to completely disable your Toolbar's search history, please click the "Options" button on your Toolbar and un-check the box next to "Drop-down search history."
If you have Google Toolbar 4.0 installed, you can clear your Toolbar's search history by starting to enter a search term in your Toolbar's search box. When your drop-down search history appears, simply click the "Clear History" link.
Please note that this only clears the search history from the Google Toolbar, which is entirely separate from the search history stored by your browser.
If you've followed these steps and information still appears in your Toolbar search box as you type, please visit http://www.google.com/support/bin/answer.py?answer=45847 for additional assistance.
CLEARING YOUR BROWSER HISTORY
Like your Google search box history, the search history that displays in the address bar on the Google homepage is stored by your browser, not by Google. Instructions appear below for clearing and/or disabling your search history in common browsers. If your browser isn't listed, we suggest consulting its online support center.
Microsoft Internet Explorer
1. Go to the "Tools" menu.
2. Select "Internet Options" and the "General" tab.
3. Click the "Clear History" button. To completely disable the history function, set "Days to keep pages in history" to zero ("0").
4. Click "OK" to exit.
MSN
If you're using the Microsoft Network (MSN), changes made in Internet Explorer (IE) will take effect in your MSN browser. After making changes in IE, you'll need to sign out of and back into MSN for the changes to take effect.
Mozilla Firefox
1. Go to the "Tools" menu.
2. Select "Options" > "Privacy" > "History."
3. Click on "Clear." To completely disable the history function, set "Remember visited pages for the last X days" to zero ("0").
4. Click "OK" to exit.
Safari
1. Click "History" on the top menu.
2. Click "Clear History" at the bottom of the menu.