How can I reduce noise caused by shift register updates?
I would attempt to lower the ESL of the high frequency signals as it is inductance that blocks high frequency.

I would double the vias of the decoupling capacitor to take the via inductance in half. I would also use an X2Y capacitor with much lower ESL (and ESR), they short high frequencies out much better than normal capacitors. I would also try an locate the capacitor closer to the ground of the shift register as the current has a longer path to take to get outside of the part, this also adds a small amount of parasitics.

Parasitic inductance on Vcc is not noticed as much as ground inductance. 

It also appears that you might have some copper that might turn into a radiator.

Make sure you have a good continuous ground plane on one of the layers (which I'm sure you do, just not apparent in your post)


Source: https://johansondielectrics.com/downloads/x2y-filter-and-decoupling-capacitors.pdf