Hi,
Its not voltage from the DMM, when in DCVolts range the only potential on the meter probes is from the supply you are measuring.
When I use the DMM to measure the voltage across one of the cap nears 74595,
Replace that capacitor and add a few more.
It sounds like with long tracks the supply is full of switching glitches the AC impedance of the meter is enough to suppress them, but I suspect a gnd loop or inadequate ground tracks.
Shorten the leads on the bypass caps you have fitted, make the cap straddle the EVERY IC, long leads can also lead (pun intended) to problems.
Other than that, place copper wire along your power tracks to the ICs and solder them in place to add to the cross-section of conductor.
ESPECIALLY your GND tracks, if it snakes around the place there is another possibility of problems.
Can you post a picture of the other side of your PCB please?
Thanks.. Tom.... 
