This guide is for eMac users to end up with Ubuntu 16.04, a custom 4.4 kernel (due to hardware limitations), and MATE desktop
Other GUI options are available, feel free to experiment with those yourself. 
I have a 2004 eMac PowerPC 7447a G4/1.25GHz (USB 2.0) PowerMac6,4 A1002 with 2GB RAM and 80GB HD.
Seeking others to test this method, and report back success/failure, or point out mistakes, make suggestions.
Any wait times longer than one minute have been noted. Your mileage may vary.
I. GETTING STARTED
Download Ubuntu Server 14.04.1 PowerPC .ISO and burn to CD.
http://old-releases.ubuntu.com/releases/14.04.1/ubuntu-14.04.1-server-powerpc.iso
II. INSTALLING THE BASE SYSTEM
Boot with the install disc
Enter at the yaboot prompt:
boot: install-powerpc video=radeonfb:1280x960-32
Enter language, country
Detect keyboard layout? Enter No
Enter language, country
wait 2 minutes
Enter host name, user name, and password
Encrypt home directory? Enter No
Is this time zone correct? Enter Yes
Unmount partitions that are in use? Arrow left to Enter Yes
Partitioning method: Arrow up to Enter Guided - use entire disk
Select disk to partition: Enter
Write changes to disk? Arrow left to Enter Yes (ERASES EVERYTHING ON HARD DRIVE)
wait 4 minutes
Enter HTTP proxy information (blank for none)
wait 2:30
How do you want to manage upgrades on this system? Enter Do not install security updates automatically
Choose software to install: Enter (without selecting anything)
wait 8:25
Remove the install disc, and press Enter to restart.
III. PARTIALLY UPGRADING (installs kernel 3.13.0-170.220)
Enter at the yaboot prompt:
boot: Linux recovery
Arrow down to Enter Enable networking, Enter Yes
Arrow down to Enter Repair broken packages
225 upgraded, 6 newly installed
Do you want to continue? Enter Y
wait 15-17 minutes
Finished, please press ENTER
Arrow down to Enter Drop to root shell prompt
Enter:
shutdown now -r
IV. DISTRIBUTION UPGRADE TO 16.04
Enter at the yaboot prompt:
boot: Linux recovery
Arrow down to Enter Enable networking, Enter Yes
Arrow down to Enter Drop to root shell prompt
Enter:
do-release-upgrade
wait 3 minutes
11 no longer supported, 9 to be removed, 132 new, 368 to be upgraded
Do you want to start the upgrade? Enter Y
wait 29-34 minutes
What do you want to do about modified configuration file 50unattended-upgrades? Arrow up to Enter install the package maintainer’s version
wait 5-9 minutes
Remove (51) obsolete packages? Enter Y
wait 1-2 minutes
Restart required Continue? Enter N
=== Command detached from window ===
=== Command terminated normally ===
press x
[screen is terminating]
enter:
apt-get purge *4.4.0 *
7 to remove
Do you want to continue? Enter Y
Enter:
exit
Enter Resume normal boot, Enter Ok
Enter user name and password at ubuntu login:
*** System restart required ***
Enter:
sudo shutdown now -r
enter password
V. BUILDING A CUSTOM KERNEL
Enter at the yaboot prompt:
boot: Linux recovery
Arrow down to Enter Enable networking, Enter Yes
Arrow down to Drop to root shell prompt, Press Enter for maintenance
Enter the following commands:
mkdir kernel
cd kernel
wget “https://cdn.kernel.org/pub/linux/kernel/v4.x/linux-4.4.183.tar.gz”
tar -xf linux-4.4.183.tar.gz
cd linux-4.4.183
cp /boot/config-3.13.0-170-powerpc-smp .config
apt-get install build-essential kernel-package libssl-dev libncurses5-dev
237 new continue? Enter Y
wait 14 minutes
What do you want to do about modified configuration file kernel-img.conf?
Enter install the package maintainer’s version
wait 1-2 minutes
Enter: make menuconfig, arrow right to enter Exit, Enter Yes to save
Enter the following commands:
make olddefconfig
make localmodconfig (press Enter to accept any defaults offered)
make localyesconfig
make all
wait 1 hour, 46 minutes
make install
shutdown now -r
VI. INSTALLING MATE (other options are available)
Enter at the yaboot prompt:
boot: Linux recovery
Arrow down to Enter Enable networking, Enter Yes
Arrow down to Enter Drop to root shell prompt, Press Enter for maintenance
Enter:
tasksel
Arrow down to [ ] Ubuntu MATE desktop, press Spacebar, Enter
wait 46 minutes
Enter:
shutdown now -r
VII. FINISHING UP
Enter at the yaboot prompt:
boot: Linux recovery
Arrow down to Enter Enable networking, Enter Yes
Arrow down to Enter Repair broken packages
wait 1:20
3 to be removed, 14 to be installed
Continue? Enter Y
wait 2-3 minutes
Remove (19) obsolete packages? Enter Y
Upgrade complete, to continue please press [ENTER]
Finished, please press ENTER
Enter Resume normal boot, Enter OK
login
System > Administration > Software & Updates > Updates Tab > Notify me of a new Ubuntu version: Never > type password > Authenticate > Close > OK
DO NOT UPDATE SOFTWARE BOUTIQUE (where it says Retrieve the latest software listings) - BREAKS IT
ALWAYS IGNORE ALL KERNEL UPDATES