It depends on what you mean by "traces". See the issue here is multifold and a lot more complicated than it seems.
Using portable browsers from a USB will not leave any personal info like, websites visited, cookies or passwords, but there might be traces as some things that need local access to work. For example, very few addons (not all) might leave a trace but if you don't use any addons, then you don't care, otherwise, you might want to search for the specific addon you use. The real question is, so what if an addon leaves a trace? As long as there is no personal info, no one will care and between you and me, someone must look specifically for what you do, in order to find anything which means they know what you are doing. Dont let paranoia hit you. Note here that this depends on the browser you use so you need to search for the specific browser you want to use.
Then again if someone wants to find out what is going on with a PC and they have software that records activity, especially via screen capture, they will see what you do no matter what (except if you boot via livecd) and any good admin worth their salt, they will either restrict USB access or restrict executing a program from USB ( for virus reasons), or have a remote control ready in case they need to access the PC quick, If now USB boot is disabled and bios password set, you can't even have a live cd boot.
So to keep it simple:
1. For simple use, pick a USB portable browser (Example: search for Opera portable browser) and execute via a USB thumb drive. All relative info will stay on USB.
2. Use a virtual machine with your own OS/browser and have it on a USB drive.
3. Use a windows PE live CD (google for more info) with a browser but you have to reboot the PC.
4. Use your own device (tablet for example) if secrecy is a top priority. You can find many cheap ones for browsing only.
Note here: Whatever you will do from USB will be slower than a disk, because USB is not meant to be used in that way and has speed issues when it's to read/write multiple small files (cookies and temp internet files), let alone run a virtual OS.
Now given that this issue is very complex and needs more details, you got enough ideas/info from this thread and you now need to research and test to see which one is better for you. The original question you asked, I believe has been answered.