You can't upgrade any PowerMac past Mac OS X 10.5.8. That means Mac OS X 10.6 Snow Leopard, 10.7 Lion, and 10.8 Mountain Lion will not work with any PowerMac.
PowerMacs with PowerPC 601 to 604 processors can all be upgraded to Mac OS 9.1, except certain Performa/Powermac 54xx/55xx/63xx/64xx machines which were limited to Mac OS 7.5.5 where X can be any number.
PowerMac G3s with a Beige case could be upgraded to Mac OS X 10.2.8. Some may require a firmware to be installed before upgrading past 10.1.5 to 10.2 or later. A third party XPostFacto software expanded some of these machine's upgrade options.
PowerMac G3s with a Blue and White case, and PowerMac G4 PCI could be upgraded to Mac OS X 10.4.11, however, could not boot externally via Firewire, nor USB any version of Mac OS X. An internal drive solution is needed to boot the installer disc of any Mac OS X. Some may require a firmware to be installed before upgrading past 10.1.5 to 10.2 or later.
Powermac G4 with up to dual 800 Mhz processor were limited by Apple to Mac OS X 10.4.11, though third party software known as LeopardAssist could force Mac OS X 10.5 to 10.5.8 to be able to be used on some of those Macs. Note with Mac OS X 10.5, Classic is gone. If your PowerMac supported booting into Mac OS 9, it would continue to be able to do so after upgrading to 10.5 as long as the Mac OS 9 drivers were left installed.
Powermac G4 with 867 Mhz, 1 Ghz and higher, and G5 machines could all be updated to 10.5.8. The PowerMac G5 is not to be confused with the Mac Pro, the user tip on upgrading to Snow Leopard explains how to tell them apart.
This user tip explains additional upgrade options.