The 3 scenarios are
1. Live USB
3. One Main OS on the computer + One Virtual OS
Live USB can be run with or without persistent storage. Let me suppose you refer to those with no persistent storage. The yes it is fairly safe.
Malware/viruses - only those that attaches to hardware - typically writes into your ROM.
Web browsing - Nothing much to say. All data wiped when reboot.
Dual Boot Systems security is as it is.
Malware/viruses - most types of viruses since the storage is persistent. Once it integrates into one of the running processes that is run on reboot you have an APT. Thus vulnerable to both software and hardware malwares.
Web browsing - Both OS will keep their records of your web browsing data probably depending on your Browser and its configuration.
Swapping over drives - This is possible if both drives are mounted. Since it is mounted then any process can access it irregardless of which OS you use. Unless you unmount according to which OS you boot on.
One Main OS on the computer + One Virtual OS is probably more secure than a dual boot.
Malware/viruses - only those that are capable of exploiting hypervisor vulnerabilities to allow interaction between the VM and its hosts. An example of such is Cloudburst created by Immunity Inc against Vmware. There exist but it is not as common since it takes quite a bit of technical knowledge to develop exploits of such. See hyperjacking
Web browsing - Likewise web browsing data probably stored depending on your Browser and its configuration.
Swapping over drives - Unlikely unless specified by the user. This is dependent on your configuration once again.
Back to Basics
The end of story is how much usability are you willing to give up for security.
Live USB and Virtual OS is definitely more secure but a great hassle.
Live USB with no persistent storage means you cannot store any of your data and softwares you need daily. Very secure but every time you log on you will need to reinstall all the softwares.
Virtual OS probably also relatively secure in the sense you can always wipe the virtualbox and reset as and when you find something is wrong. However since it still possess storage to softwares and their configuration it is still possible to attract software exploits within the VM. Its harm is USUALLY limited to the VM though as mentioned above.