Put some decoupling capacitors on the shift register.http://www.thebox.myzen.co.uk/Tutorial/De-coupling.html
I would put a place for a 10uF capacitor on each board where the power goes in. However you will probably not have to fit one on every board just one on the first in the chain and the last.
Is the pinout for the display correct? I've used them the past and it was not the same as yours but then they may have been different displays.I've attached a portion of my schematic, note that there are two anodes.The PCB looks OK, quite a few odd angles on traces but that won't break the bank. I seem to recall that the spacing on the rows of pins was not a multiple of .1", maybe not but worth checking.Also with the schematic, you should at the least add the pin names to the chips, nobody is going to search the data sheets to figure it out, and it's also harder for you to debug. And move the pins so the drawing flows logically, it's very difficult to follow a "real life" pin layout with dozens of parallel and crossed over traces.
Might want to check my project out: http://arduino.cc/forum/index.php/topic,71211.msg530649.html#msg530649The onboard Arduino and associated components can be left out; focusing on the shift registers you could reduce your board component count to just a single IC, a resistor, and a decoupling cap. STP16C596 and TLC5916 are pin compatible DIP versions of the TLC5925. There are also 8 output ICs like these.The code to drive it is also available and easily modified for any number of segments that you want to drive.