Is it possible to drive 120 solenoids from 120 outputs of shift register 74HC595?
First, you do need flyback diodes across the coils, or zeners + diodes if you need the solenoids to switch as fast as possible. 

If the chips are getting extremely hot, that's a strong indication of latch-up. 

This is not an uncommon situation when you combine large currents and sub-optimal layout. Chips that are connected to each other have protection diodes on the inputs and if one chip's ground bounces by more than a diode drop or two you can have large currents flowing between CMOS output and input, through the protection diodes. If that current exceeds some limit (probably in the range of 100mA for modern parts) the CMOS chip will latch up and may be destroyed if there is enough power supply current available. 

The solution is to separate the high and low current paths so that the solenoid return current from the TIP emitters does not flow through the grounds on the shift registers. Connect them together at one point, usually the power supply return. The 1K resistors will prevent anything too dastardly happening if your layout isn't the greatest, but try to keep the return on the transistors physically short (low inductance). Fortunately the TIP transistors are very slow to switch so the problem isn't magnified. 

If you do try MOSFETs, keep the resistors, in fact you can increase them to a few K to slow the switching.