I am not injecting opinion here. This does not reflect my overall opinion or preference but is specifically answering the question. I have no desire to debate preferences of distributions.
Ubuntu Fedora CentOS
which one (and which version) would be the most stable?
Between these choices, CentOS is definitely the most stable. Stable in the sense that the packages included are mature and tested, which often does not included the latest major version of the software. CentOS is based on RedHat Enterprise Linux, which specifically applies standards desired to enable the highest level of stability for professional environments. This includes a 7-year release cycle.
Fedora is the opposite of CentOS in the sense that it includes the latest major versions of software as well as "new" features that are being tested before being introduced to the Enterprise version of RedHat. Fedora's life cycle is shorter and has a completely different approach.
Ubuntu is newer to the industry and most commercial support is for RedHat Enterprise Linux. However, Ubuntu does offer a version (LTS) with a longer release cycle. Still, this cycle is shorter at 5-years. It could be argued that Debian is a better choice, as it is more mature and Ubuntu is based on it. Ubuntu was created with a focus on usability. With this, their focus appears to have historically been on end-user features.