Firefox is for a SEO researcher not only a simple browser. It is a SEO tool which is making your optimization work easier, your browsing will be enriched with small SEO options like watching your competitors. If you want to use a desktop application to watch parameters of a website optimization you can useWebsite Popularity or you can use your Firefox browser:

Here is bellow my list of Firefox plugins which are helping me on the SEO work or on checking some parameters for a website, for bloging etc:

Firebug

Firebug integrates with Firefox to put a wealth of web development tools at your fingertips while you browse. You can edit, debug, and monitor CSS, HTML, and JavaScript live in any web page.

ScreenGrab

Screengrab! is an extension for Firefox that makes it easy to save a web-page as an image. With it, you can save anything that you can see in a browser window – from a small selection, to a complete page.

Google UnPersonalized

This plugin prevents you from seeing personalized search results, when searching from your browser’s search box. In Firefox, this plugin also adds suggest functionality.

InFormEnter

InFormEnter adds a small, clickable icon next to every input field in a web form, from where you can select the item to be inserted – no typing required. You can configure it to display your frequently used information such as name, email, address…

User Agent Switcher

Adds a menu and a toolbar button to switch the user agent of the browser.

Search Status

SearchStatus is a toolbar extension for Firefox hat allows you to see how any and every website in the world is performing. Designed for the highly specialised needs of search engine marketers, this toolbar provides extensive search-related information about a site, all conveniently displayed in one discreet and compact toolbar.

TwitBin

TwitBin is an extension for firefox that brings the power of twitter right in your browser.

Seo Quake

Seoquake is a powerful tool, aimed at helping web masters who deal with search engine optimization and internet promotion of web sites. Seoquake allows user to obtain and investigate many important SEO parameters of the internet project under study on the fly.