That's interesting, so you can flip the relays all you like without trouble as long as the 24DC supply isn't connected? If that's true then your problem presumably isn't the typical inductive kick from the relay coil. It looks like your relay board has stuff on it which is presumably drivers and snubbers so let's assume all of that is adequate to the job.
So, if it's inductive kick from the valve solenoid it's being coupled all the way from there, back through the 24DC supply to the outlet, then forward through the USB supply to your shift register, which is impressive! But not implausible.
Anyway, three places I'd add some stuff:
- The main thing you need is a snubber network across the valve solenoid coil itself, ideally physically close to the valve (you want to minimize the area of the loop formed by the valve coil - wiring - snubber). Something as simple as a freewheel/clamping diode would probably help a lot. This will also improve the lifespan of your relay contacts which are probably arcing a little.
- Small decoupling cap on your breadboard, say 0.1µF on the power supply rails, to keep your logic happy.
- Larger decoupling cap on your 24VDC rails (the bus on the left), just to eat any transients the snubber doesn't deal with. Maybe 1-10µF or so?