About Opera Portable: No traces, except Flash cookies
Opera 12.xx will save all files inside its profile. If you choose Portable mode, that folder is places inside your installation folder whereas this folder should be somewhere on your USB stick.
Some examples where Opera stores temporary files (on your USB stick)
There won't be any traces left if you remove the USB stick.
But: In normal cases Adobe Flash Player is installed on the main system and Opera will use this plugin when installed. As soon as you visit a Flash-based website, the Flash plugin will save cookies to:
and this will leave traces. Read here fore other browsers and there Flash cookie locations