Shift register m74hc165 connection

How is the shift-register wired? Do you have a 0.1uF bypass capacitor across the power leads close to the IC package?

It sounds to me like a noise issue, particularly since it's only triggered when you have a load on the relays. The fact that it results in the shift-register register-state getting reset makes me think it's a power issue.

Also, how are you wiring the shift-register.
With a 74HC595, you need to:

• Tie the two register clocks together (Pins 11 and 12)
• Pull the master-reset pin high (tie pin 10 to VCC)
• Pull the output-enable line low (tie pin 13 to ground)

Lastly, you need a 0.1 uF bypass capacitor between pin 16 (Vcc) and pin 8 (Gnd).

If I understand your question correctly, it is important to understand that the device works in a couple separate stages.

1. The parallel inputs. These have latches that must be signaled to read the inputs and internally hold their state. Once latched, the signal on the input pins can change but the internal signals will be stable. (RCK)
2. The next step is where these internal signals are copied into the shift register. (\$\overline{\small \text{SLOAD}}\$)
3. The last step is shifting out the bits in the shift register (SCK)

If I understand the truth table and timing diagram in the datasheet correctly then the correct order of steps is the following:

1. Initialize the signals as follows RCK = low; \$\overline{\small \text{SLOAD}}\$ = high; \$\overline{\small \text{OE}}\$ = low; SCK = low;
2. Apply parallel signals to the inputs A~H
3. Clock the parallel signals into the latches by driving RCK high. RCK triggers on a rising edge. Drive RCK low, its inactive state.
4. Copy the contents of the input latches into the shift register by driving \$\overline{\small \text{SLOAD}}\$ low. \$\overline{\small \text{SLOAD}}\$ is active low. Drive \$\overline{\small \text{SLOAD}}\$ high again, its inactive state.
5. Read pin Q_H on your MCU
6. Apply a clock pulse on SCK to shift the data out on pin Q_H. SCK triggers on a rising edge. Drive it high, then low.
7. Repeat last 2 steps for as many inputs as you are interested in.