Most of the popular distros are stable unless you stray out of their subscribed repositories. There are many ways of creating instability while running a particular distro if you ignore their manual and install things generically by compiling code on your own from the developers website.
For example, in Fedora, if you decide to use the Nvidia proprietary drivers and don't follow the instructions that are in the forum, and, choose to use the .run file from the Nvidia website, you're certain to wind up with a broken system. Or, another example, is downloading OpenOffice or LibreOffice from their website instead of installing it from the Fedora repo, you can also run into problems. I'm not saying folks shouldn't experiment, but just realize that if you don't RTFM, occasionally you'll get bitten.
Sometimes when main features go through major revisions, such as, the switch from KDE3 to KDE4, or, the switch from Gnome2 to Gnome3, or, even the introduction of Unity by Ubuntu, the stability of the version that features that change drops until all the bugs are ironed out, but, these changes don't happen very often, so, saying that there's a problem with stability in Linux distros isn't a correct assessment.
As a matter of practice I use distros that skirt the bleeding edge so I can test and file bugs, and, even though I run distros based Debian Sid or run Fedora with the testing repos enabled, I've had very few instances of instability in the years I've been using Linux.
It's all a matter of how you use you're system as to whether it remains stable or not. Most distros take great care in not releasing unstable versions of apps into their main repositories.