To provide a somewhat definitive cap statement to this question, as of November 2021 the MacPro3,1 (2008) is really beyond obsolete in the modern macOS world.
Now that being said as someone who squeezed every ounce of usefulness out of one, I can state that you can pull one seemingly as modern as Catalina if you use dosdude's macOS Patcher.
The official end-of-line is El Capitan (10.11). This version of macOS will happily install on a 2008 MacPro.
The reasonable end-of-line is High Sierra (10.13). "Reasonable" in that you don't need to do much other than upgrade the Bluetooth module to make it work and the performance is acceptable. As well, you can get away with most video cards at this point.
The UNreasonable end-of-line is Catalina (10.15). This is "UNreasonable" because video performance is heavily impacted unless you get a top-of-the-line aftermarket video card (e.g. the 680GTX), but then also DosDude's patcher is doing a lot of things to keep this rig going at that point.
Big Sur presumably does not work on the 2008 machine.
In my own experience, I found that Mojave (10.14) was where it started to get a bit painful. I was able to get along with a Radeon 270X GPU, but there were a number of places in the OS where things were just really slow and you could tell that the MacPro3,1 was struggling. It was at that time that I realized 12 years was pretty good for the box and it was time to retire it instead of throwing any more cash into it.
That being said, it does work very well as a Windows 10 box if you want to get a few more years out of it!