If it updates itself, it can leave temporary install files in the "%windir%\temp" and or the "%temp%" folders. If you run Flash, it can leave your tracks in the "%AppData%\Macromedia" folder. If you download a file, it could be left on the Windows computer if you don't choose to download to the USB drive.
When you use Opera on the Windows install, DNS entries will be left behind. You can see them with ipconfig /displaydns in a command prompt and clear them with ipconfig /flushdns.
That should be about it though. You'd have to test if the USB install accidentally creates anything in the %AppData%\Opera Software" and "%LocalAppData%\Opera Software" folders on the Windows computer just to make sure, but it shouldn't.
Pretty much though, your Opera data and settings and passwords etc. will not leak.