I prefer Xen over anything else.
As previously said, Xen provides the stability by being a Premium Service.
KVM is just as HVM, with the difference that in KVM you can oversell RAM. In Xen you can't. (This means that with Xen you pay what you get, that's why it's more expensive).
The question that's driving this market is Price since the new users tend to get the cheapest plans possible.
Since you can oversell RAM in KVM/OpenVZ, the price per GB drops. In OpenVZ this is even more critical, because you can also oversell HDD. This has to do with the arquitecture of each hypervisor.
So, trying to mesh things up:
If you want to have a starter environment where you want to learn how to handle Linux, OpenVZ is your thing! It's cheap, it's simple, it works! Don't expect stability/performance on *very* cheap plans, though. But since it's for learning purposes, I think it's fine.
If you want to be on the same level (starter environment, still linux) and you want/need stability, you should go with Xen PV. This is *almost* the same as OpenVZ, but the resources are dedicated to you.
If you need a full Virtualization but you do not need performance, you should go with KVM (it's cheaper).
If you need full Virtualization, but you need stability, you should go with Xen (more expensive).
In sum, this is my personal opinion:
Stability + Performance = not cheap (Xen)
Cheap + Performance = not stable (OpenVZ)
Stability + Cheap = not performant (KVM)