An advanced attacker could compromise a hypervisor through emulated device drivers more easily than through clipboard handling. The way a hypervisor works is by providing a new execution environment for a guest, and exposing emulated hardware. When a driver in the guest tries to access the virtual hardware by, for example, reading and writing to PCI BARs to interact with PCI devices, the hypervisor's emulated device software will hear the request, temporarily pause the running guest, and emulate in software what the actual hardware would have done before unpausing the guest and allowing it to see the results. This allows the guest to be effectively unaware of its underlying virtual environment. When you think of the emulated devices as acting merely as a proxy, implementing a complex and often poorly documented protocol between the guest software (which is designed to talk to real hardware) and the host hardware (which is capable of listening to software), it's easier to see how a bug in the parsing of such complex data structures could result in a security breach.
This is a look into some of the security caveats of a hypervisor, specifically QEMU with KVM, though the general concepts should apply to pretty much any hypervisor software.
Note that most of the time, exploiting a bug in emulated devices requires the attacker to have privileged access to the guest. They have to completely compromise the guest before they are even allowed to interact with the emulated hardware. If the guest itself is sufficiently secure, then the hypervisor will likely be, as well.
The full version of the famous quote from Theo de Raadt, founder of OpenBSD:
x86 virtualization is about basically placing another nearly full
kernel, full of new bugs, on top of a nasty x86 architecture which
barely has correct page protection. Then running your operating
system on the other side of this brand new pile of shit.
You are absolutely deluded, if not stupid, if you think that a
worldwide collection of software engineers who can't write operating
systems or applications without security holes, can then turn around
and suddenly write virtualization layers without security holes.
You've seen something on the shelf, and it has all sorts of pretty
colours, and you've bought it.
That's all x86 virtualization is.
In the end though, the chances that a "dodgy link" will contain a hypervisor-breaking 0day are pretty remote. If anything, it'll have an exploit for an old version of Flash, used to download botnet software.