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At first this sounded like it might be a classic case of forgetting the flyback catch diode across the relay coil.  However, now that you've posted a schematic we can see that a diode is built into the driver chip for each output.

That leaves two possibilities I can think of, power overload and ground bounce.

These are not great drivers, dropping 1.6 V.  You didn't say how much current the relay coils draw, so maybe the drivers are over-heating.  Individual outputs are rated for 500 mA, but note that the whole package is only rated for 1.4 W at 25°C, and less at higher temperatures.  1.4 W spread over 8 drivers is only 175 mW per driver if all 8 can be on simultaneously.  With the large 1.6 V drop, that means the actual current limit per driver is 109 mA.  Check that your relays draw less than that.

Another possible problem is ground bounce.  Since these relays are switching motors, which are presumably high-current loads, you have to carefully consider where those currents flow.  If the ground of the driving circuitry and the motors are tied together, then you should avoid having the motor return currents flow thru the ground of the driving circuitry or the wires connecting it back to the power supply.  The motor current times the impedance of the connection back to the 9 V supply causes a voltage offset, which could be large enough to damage your relay drivers.  Your scope trace with the glitch when the motor turns off is a hint that this may be happening.

Induction from the motor's magnetic field to the relay coil is theoretically possible, but unlikely unless these are really big motors.