Amie

This widget replaces the standard Blogroll widget, and adds the following features:
<ul>
<li>API compatibility, so the widget matches the theme</li>
<li>Respects options set in the Links Manager Admin Panel</li>
</ul>

WPW-Linkslist

A flexible replacement for the standard links widget, for wordpress 2.0.x only.

Enej Bajgorić

ctlt-dev

ubcdev

Aleksandar Arsovski

This plugin allows users to display their WordPress links by category through custom fields.
The following gives a step-by-step setup overview:
<ol>
<li>Enable the plugin (check the installation tab)</li>
<li>Go to the page/post where you’d like to place your links/blogroll</li>
<li>Make sure that Custom Fields are visible (on the edit page or post page click on the “Screen Options” in the top right and make sure “Custom Fields” is checked off)</li>
<li>Under CUstom fields click on the enter new link</li>
<li>Enter an ID in the “Name” field – the plugin uses the ID “display_links” as a default</li>
<li>Enter the category name of the links you’d like to display in the “Value” field</li>
<li>Click on the “Add Custom Field” button</li>
<li>Enter the following shortcode into the post content area or the widget area you prefer: <code>[links_by_cat field_id="(ID previously entered in Name field goes here)"]</code>
<ul>
<li>Note that if you used “display_links” in the Name field, you do not need to enter the “field_id” parameter in the shortcode and can simply do: <code>[links_by_cat]</code></li>
</ul>
</li>
</ol>
The shortcode also supports many other parameters for ordering and structuring your links. All parameters on the list (except “category”) found on the following page are available: https://codex.wordpress.org/Template_Tags/wp_list_bookmarks#Parameters
Other parameter example:
<pre><code>[links_by_cat orderby="rating"] -- This is an example using the orderby parameter from the list. The links will now be ordered by their rating.
</code></pre>
<h4>Added in 1.0.1:</h4>
The ability to display all categories has been added to the shortcode. Simply add: 
 [links_by_cat display_all]
Many bug fixes have also been introduced in version 1.0.1 as well, so please upgrade.
Currently, only one category is supported per Custom Field entry; however, I am looking to update the plugin soon so that it supports multiple categories.

Display Links by Category

A simple shortcode plugin for displaying links by category through custom fields.

WPW-Linkslist vs Display Links by Category