How do you clear your browsing history?
Get Sandboxie
Install a RAM disk
Point the Sandboxie container folder to the RAM disk.
Run your web browser sandboxed.
Restart your computer and all history is history, gone for good. Really, truly, and beyond recovery.
Sandboxie is freeware. However, if you register the software for even more options, you'll get free lifetime support and updates, and you may install the software your license on as many computers you own.
(This works for Windows only.)
Personally, I see from Molly's comments, you are concerned about the whole infrastructure - logs etc.
The first thing,
Do not try at work or any machine you do not own. You simply do not know what monitoring tools are installed, there could be keyloggers, or ANYTHING that sends everything to a remote location completely outside of your control.
Next, as for your actual machine, you really want to run all the usual tools such as CCleaner to clear out your local machine from every day objects. This should work well, but there is always a remote chance someone can recover your files - You may want to look in to either trashing your hard drive, or doing a complete wipe with something like DBAN.
Lastly, infrastructure - Logs are kept at many different levels. Your ISP will have various logs that can contain information on sites visited, your routers DNS can have a cache (but it is usually wiped after a few hours)... Your best bet is to use a web based proxy service or even better TOR.
However, if someone really wants to trace what you have been doing, they will try to find a way - ask ISP for sites visited if they keep logs... if they see a proxy server, they can ask them for logs... Your best hope is an ISP that doesn't keep many logs or perhaps using a VPN type service in a foreign country.
All this being said, do not do anything illegal!... That is the best way to not get into trouble for browsing!
Ignoring secure-delete (which probably isn't that important) for a minute, the most notable history trace you'll gather that isn't removed by Firefox's “Clear Recent History” option is Flash data storage. The storage control feature of Flash is unfortunately separate and has unusable controls which don't even delete the full history.
On Linux you can remove these properly by deleting .adobe and .macromedia in your home directory. (It's a problem on Windows too, similarly requiring futzing around in Application Data.) A good longer-term strategy is to use Flashblock, to stop every untrustted site and ad network dropping unwanted cookie-like storage and history traces.