The sidebar search input is stored in $:/temp/search,
and the $:/AdvancedSearch input is stored in $:/temp/advancedsearch
The filter that determines what is saved in the file is defined in $:/core/save/all, which contains:
\import [[$:/core/ui/PageMacros]] [all[shadows+tiddlers]tag[$:/tags/Macro]!has[draft.of]]
\define saveTiddlerFilter()
[is[tiddler]] -[prefix[$:/state/popup/]] -[[$:/HistoryList]] -[[$:/boot/boot.css]] -[type[application/javascript]library[yes]] -[[$:/boot/boot.js]] -[[$:/boot/bootprefix.js]] +[sort[title]] $(publishFilter)$
\end
{{$:/core/templates/tiddlywiki5.html}}
Note the $(publishFilter)$ reference in the saveTiddlerFilter() definition. This macro variable can be used to list additional tiddlers that should be excluded when saving the file.
Thus, to prevent the search inputs from being saved all you need to do is to create a global macro tiddler (e.g., "MyPublishFilter", tagged with "$:/tags/Macro") like this:
\define publishFilter() -[[$:/temp/search]] -[[$:/temp/advancedsearch]]
When you save your document, those two tiddlers will no longer be included. As a result, when you reload the file, the search inputs will default to blank, as desired.
enjoy,
-e
Eric Shulman
TiddlyTools.com: "Small Tools for Big Ideas!" (tm)