Highest Apple-supported macOS versions:
Mac Pro 2008 (3,1) - El Capitan (OS X 10.11.6)
Mac Pro 2009 (4,1) - El Capitan (OS X 10.11.6)
Mac Pro 2010-2012 (5,1) - Mojave (OS X 10.14.6) with Metal-compatible GPU (or 10.13 without metal GPU)
The 4,1 can be flashed with 5,1 firmware to make it a 5,1 for all practical purposes.
Unofficially, you can use the dosDude patchers to upgrade a 3,1 or 4,1 to Sierra (10.12), High Sierra (10.13), Mojave (10.14), or Catalina (10.15). It will take a 5,1 to Catalina. Note that you really should use a Metal-compatible GPU for Mojave or Catalina.
If you go to Mojave or higher with a 5,1 be sure to stop at Mojave at least temporarily, to update to the latest 5,1 firmware. Later macOS versions (and some peripherals, like M.2 devices) will work a whole lot better (or at all) with the latest 5,1 firmware. See this thread for details.
Unoffically, you can use OpenCore Legacy Patcher (OCLP) to update a 3,1 or 4,1 or 5,1 to Big Sur (11) or Monterey (12). OCLP is able to make a non-Metal GPU work, but you need a metal GPU to do the firmware update in Mojave. So you need at least one Metal GPU for 4,1 or 5,1 updates.
Metal-compatible GPUs: AMD (Radeon HD 7xxx or higher), nVidia (GTX 6xx/7xx or 9xx/10xx with web drivers). This is only a general guideline - check the GPU stickies for more specific info. Also, it's best to avoid nVidia cards, especially for recent macOS versions. Apple dropped support for them long enough ago, that the cards have aged and drivers are badly outdated.
OCLP will go higher, to Sequoia currently, and Tahoe is in the works. However, there are GPU restrictions (no AMD navi cards) for a 2008-2012 Mac Pro. The lack of AVX opcodes in our CPUs also causes some software to fail - you need software versions compiled without AVX codes. If the software says it will work in Monterey or earlier, it will work in our OCLP installs of Ventura through Sequoia. Presumably Tahoe as well in the future.
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