I am running Ubuntu server 12.04 and my fresh installation is using the
3.2.0-60-generic kernel, however when I go to https://www.kernel.org/ it says the latest stable kernel is only 3.14
I don't want to run into problems down the line.
The kernel released with 12.04 server at its outset was v3.2.
Canonical have committed to supporting v3.2 of the kernel:
What this means is that the kernel v3.2 will receive security and other stability fixes throughout the 5 years of the support period.
Newer kernels are available for each point release - these are called enablement stacks and are primarily for newer hardware. You dont need to install these if kernel 3.2 is just running fine on your hardware.
Obviously, if you need to add newer hardware cards to your server you should first check that the cards work with the kernel 3.2 series. Likewise, if you image the server to install on newer hardware, then you should similarly check 3.2 compatibility.
I think the OP might be (or was) under the mistaken impression that 3.2 is newer than 3.14 because he's viewing 3.2 > 3.14 in a decimal sense instead of thinking about three separate numbers. He said " the latest stable kernel is only 3.14," which makes me think his concern is that 3.2 is too "bleeding edge." He's obviously good to use 3.2 in either case as was rightly pointed out, but I wanted to make sure he understood the versioning system for future concerns.
Yes, the 3.2.0 kernel is absolutely safe to use.
Linux distributions (Ubuntu 12.04, Ubuntu 13.10, CentOS 6.5, etc.) will typically choose, and stay at, a base kernel version in order to maintain security, stability, and other features that might cause issues if moving to a newer kernel.
The newer kernel versions usually have cool, "bleeding-edge" features (performance improvements, updated drivers, etc.), but with the sacrifice of being less stable and less tested. The 3.2.0 kernel has been tested and vetted by dozens of organizations and individuals that have made absolutely sure that it is stable enough to host "production-ready" applications and environments.
To compare, CentOS 6.5 runs kernel version 2.6.32-431, which multiple enterprise organizations use to host mission-critical applications and devices.