zackdesign

Turn your WordPress post categories into a search powerhouse! This plugin gives you the ability to provide your users with a widget full of dropdowns based upon parent categories and their sub-categories.
A new function has been added for release 2.2 – you can now set up an AJAX chained select box. For example, if you have multiple countries, and multiple states for each country, you can get a user to select a country. 
Upon selecting the country, the plugin looks for any states under that country, and then shows the dropdown box for the states under that country beneath the country dropdown.
Multiple subcategories spanning multiple parent categories (though multiple categories inside a single parent cannot be chosen) are chosen and intelligently filter your posts.
Please note that this plugin, while it is a widget, can be used on a sidebar you define anywhere within your theme that you like. Think of it as a ‘block’ area rather than a ‘sidebar’.
Sample Application:
Real Estate – set up parent categories such as ‘Bedrooms’, ‘Bathrooms’, ‘Price Range’, etc. Then put sub-categories beneath them. These are the ones that are searched.
Create a new post and tick all applicable sub-categories. Users should now be able to search for your post under all those categories!
Features:
<ul>
<li>Inclusive category search</li>
<li>Exclusive category search</li>
<li>Ordering of results</li>
<li>Automatically checks for the ‘uncategorized’ category and hides it</li>
<li>Generates complete URL </li>
<li>Now does a proper URL-Rewrite if permalinks are enabled</li>
<li>Pagination</li>
<li>If search is pressed, allows you to decide whether or not to show all posts (or a post category ID) or none</li>
<li>Exclude certain categories</li>
<li>Tested with All-in-one-SEO, WP Smart Sort, and sCategory Permalink. Now plays nice with all four! Required a bit of a logic rewrite, thus the 2.0</li>
<li>Supports the new WP 2.8+ widget class structure and as a result is now available in as many instances as needed</li>
<li>2-level AJAX chained select options</li>
</ul>
Please be aware that I’ll only be updating this if I need to or if I’m paid to. Feel free to come on board and contribute!

Multiple Category Selection Widget

This widget makes it easy for your site users to drill down into your Wordpress post categories.

Murali

WordPress press widget to list category and subcategories of posts/custom taxonomies of custom post types.
Features
<ol>
<li>
It allows users to add custom icons for category or for subcategory or for custom taxonomies
</li>
<li>
It lists categories and subcategories in user-friendly horizontal pattern. Please refer screenshots.
</li>
<li>
Category icons can be edited/removed/update with new icons under edit category or under edit custom taxonomy admin screen
</li>
<li>
Admin settings of this widget has facility control display on/off for category icons
</li>
<li>
Admin settings of this wiget has facility to show/hide count of posts in categories
</li>
<li>
Admin settings of this widget has facility to show/hide empty categories in widget
</li>
<li>
Title of widget can be customized in widget settings
</li>
<li>
Compatible with blocks under widgets section
</li>
</ol>
<h4>My Links</h4>
<ul>
<li><a href="https://www.muraliwebworld.com/groups/wordpress-plugins-by-muralidharan-indiacitys-com-technologies/forum/topic/category-sub-category-widget-for-wordpress-websites-using-php-javascript-jquer/" rel="nofollow ugc">Plugin homepage</a></li>
<li><a href="https://wordpress.org/plugins/push-notification-for-post-and-buddypress/" rel="ugc">Push notification plugin for WordPress,BuddyPress and for Mobile apps</a></li>
<li>Twitter @<a href="https://twitter.com/indiacitys" rel="nofollow ugc">indiacitys</a></li>
<li>Facebook <a href="https://www.facebook.com/indiacitys" rel="nofollow ugc">indiacitys</a></li>
</ul>

Category and Subcategory List Widget

This widget allows to add/update icons for category or icon for custom taxonomy. It lists Categories in horizontal menu pattern.

Multiple Category Selection Widget vs Category and Subcategory List Widget