Here are some of my favorites. One of them (Counter) behaves strangely. It doesn't appear to be stable, but it takes a long, long time to change appreciably. Another one (Sewing Machine) seems semi-stable in that it seems to settle around the same kind of almost-but-not-quite-chaotic pattern.
Counter: http://wry.me/hacking/Turing-Drawings/#5,3,1,1,0,3,1,3,2,1,2...
This one seems to behave like some kind of adder. Even at the highest speed setting it takes a long time to see much change, but the black lines seem to grow and move similar to how a binary adder operates.
Warpspeed http://wry.me/hacking/Turing-Drawings/#2,11,1,5,1,0,1,1,1,5,...
City Corner http://wry.me/hacking/Turing-Drawings/#2,11,1,10,0,1,5,2,0,8...
Coloring Inside the Lines http://wry.me/hacking/Turing-Drawings/#2,11,0,10,2,0,4,3,0,2...
Sewing Machine http://wry.me/hacking/Turing-Drawings/#2,11,1,7,3,0,2,0,0,3,... Seems semi-stable?
Folding Paper Trick http://wry.me/hacking/Turing-Drawings/#2,11,1,4,1,0,5,3,0,10...
Stable http://wry.me/hacking/Turing-Drawings/#5,3,3,1,0,3,1,0,3,1,0...
Counter: http://wry.me/hacking/Turing-Drawings/#5,3,1,1,0,3,1,3,2,1,2... This one seems to behave like some kind of adder. Even at the highest speed setting it takes a long time to see much change, but the black lines seem to grow and move similar to how a binary adder operates.
Spilled Drink http://wry.me/hacking/Turing-Drawings/#13,21,10,11,1,0,7,0,4...
Donkey Kong http://wry.me/hacking/Turing-Drawings/#13,21,0,17,3,11,4,0,0...