wokamoto

You can choose (child) categories and see the posts belong to the chosen category with AJAX.
When the category you choose from the list box has child categories, the child categories will be shown in a new list box. 
Clicking the “Search” button, the list of posts belong to the bottom category which you choose will be displayed under the “Search” button.

Simple Category Search

You can choose (child) categories and see the posts belong to the chosen category with AJAX.

livecanvas

The Purpose: allowing your users to search differently
Ajaxed search engines are all over the place ? literally. 
Whenever a large amount of items is available, be it houses for rent 
or computer spare parts, providing your users with a fast, structured, faceted search 
can be one thousand times more efficient, rather than letting your visitors sink into hundreds of web pages, 
hoping they?ll find what they need ? almost by chance. 
This tool does not replace or try to extend the build in textual search feature of WordPress; 
it aims to provide a quick bootstrap to an ajaxed search engine, demonstrating how easy it can be to turn 
your website into a powerful search tool, without installing a huge and bloated code.
It is easy to hack and to extend at your will. 
You can read here the original article about why <a href="http://www.dopewp.com/searchy/" rel="nofollow ugc">Searchy</a> was built.
<h3>Main Features</h3>
Let your users search combining different criterias: Filter by name, category, tag, or custom field (=1). 
Sort results by date, name or custom field (numeric)
<h3>Typical usage</h3>
Searchy is a customizable search engine composed of two elements: the search filters – typically a filters column, and the search results.
Use the searchy_filter shortcode or the built-in widget to show the filters column. 
Place the searchy_results shortcode in a WordPress page to show the sort bar and the search results. 
Of course those go in square brackets in a chosen Page. 
You can place them one after the other, having searchy_filter and searchy_results close, 
Or get fancy building your custom layout, placing some do_shortcode calls wherever you want.

Searchy

An AJAX Search Engine allowing different criterias for sorting and filtering your posts. Use via simple shortcodes and few settings.

Simple Category Search vs Searchy