Controlling 12 Solenoids from Arduino / ShiftOut

No that is very misleading.
The data sheet says:-

Eight Power DMOS-Transistor Outputs of 150-mA Continuous Current
500-mA Typical Current-Limiting Capability

The 500mA is for pulsed current where .... Pulse duration ? 100 ?s and duty cycle ? 2%.
So nowhere remotely long enough to turn on a solenoid.

Given that the OP is using a solenoid ...... Rated Current: 0.58A
Then the TPIC6C595 is not suitable to drive it.