In my experience, Ubuntu 12.04 is less stable than Windows 8. Ubuntu is more likely to freeze, crash, or otherwise act badly than Windows.
When I say this, people point out that Ubuntu is not the Linux kernel and that the problems I see come from the GUI or from Ubuntu itself. I can believe that.
So Linux is really stable when you don’t run it on a desktop. But the same is true of Windows. If you set up a Windows server and treat it like a server — don’t install random consumer software, don’t browse the web on it, don’t play games on it, etc. — then Windows is really stable too.
When people compare the stability of Linux and Windows, they may be biased a couple ways. First, Linux is more often deployed on servers and Windows more often on desktops. So they may unintentionally be comparing Linux servers to Windows desktops. Second, they may be thinking that Linux users’ computers are more stable than Windows users’ computers, which is probably true. Linux users typically know more about computers than Windows users. They are more able to avoid problems and more able to fix them when they occur.
I suspect the most important factors for a computer’s stability are how it is being used and who is using it, not what OS it is running.