It clears all search history after you close LW.
You can get rid of individual search results by clicking on the red square with an x in it at the top corner.
If you're refering to autocomplete where when you add the 1st one or 2 letters of a previous search it automatically fills out the remainder. Then if you don't like this go to: LW's menu Tools>Options>View>Autocomplete & untick the option box & press the Apply button. For OSX that's LW's Prefs>View>Autocomplete & Mac OS9 that's Options>View>Autocomplete |