Flyback diode mesa 7i95t

The issue here is that the relays are switching a 120VAC solenoid.

Because the solenoid is an inductive load, this will cause massive EMI unless
there is contact arc suppression across the relay contacts. This can be done
with a bipolar TVS diode, a MOV or an RC arc suppressor.

Some do some don't, you'll need to check the data sheets.
The role of the diode is to protect the driving circuit from the reverse voltage generated from the collapse of the magnetic coil. No flyback will eventually cause component failure, a far bigger issue than EMI.
I'd take PCW's advice.
Another thing I would look at is the routing of your wiring and ensuring physical seperation of control and AC, if you haven't already taken care of this issue.
If you do use optos make sure that the input side and output side are powered by seperate PSUs and the ground are not connected, it aggravates me when I see opto circuits with the input and output sides connected to a common ground.

From what I can see, the relay board in question uses an optoisolator to drive a relay via a tiny SMD transistor, which could be a FET.

Also, take advantage of those optoisolators (superfluous though they are) and power the relay board with a totally separate 12V DC supply.

The power supply for the relays does not need to be that powerful.

Using the 12v rail from your PC negates the protection the optos are used for.

I suggest to forget all the BS you've learnt from the Hipster Maker crowd and read & learn by someone that knows his stuff.

This is a good start, written by the guy that designed the TEC-1 sbc back in '83 or '84:
www.talkingelectronics.com/te_interactive_index.html

What if you filter the power?

(no big caps tho)

I power the cooling fans of my drivers of my PC's 12V rail.

It is perhaps a bit dangerous, but it does make it possible without a second supply.

What about if the relay board and mesa share a common power supply, but have totally separate power wires (connecting only at the PSU itself)?

The function of an optocoupler is to completely isolate one side of a circuit from another. Hence the large no copper regions under the device.
No amount of filtering, hoping nor thinking will provide this if the same supply is used or if 2 different supplied are used and their grounds connected.
Pickup a book, read & learn.
Eventually your mis advice will end with someone with less knowledge and experience than yourself having a bad time.
Those unseen electrons moving around can bite.