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I've started switching all to bluetooth (printer, keyboard, mouse), wifi, and HDMI (usb as a last resort) which has cleaned up my cabling somewhat. Although right now I'm stuck on ethernet because my wifi dongle isn't working with Ubuntu this time (was last time *shrug*)
I really must do something about it all, but ... the trouble is that I used to have a UPS which had IEC female, so when it died I used standard IEC lead to connect the IEC males that went to the kit instead of ripping it all out and wiring it properly. So I have a rats-nest of mains, another of CAT5, another of USB, ...
And a sh*t load of dust that I don't dare go near because I know something will stop working if I do, and it's dark down there ...
"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony
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Split loom tubing / corrugated sleeving . McMaster Carr in the US, or your local Industrial electrical supply. Spiral wrap ( sleeving ) works very well, but at 2 turns per inch on small diameters it's slow. The zipper stuff can be nice.
But, I'd just use velcro, 3/8" wide roll cut the ends on a diagonal - it makes it easier to wrap and to remove. And you get to tie to legs... to hold things in place. ( We cheated and used large binder clips to dress cables to the metal desk. )
I am curious about which kinds of cables you are troubled with.
"In the old days" we had no sort of buses: The printer had its LPT cable, the modem its COM cable, the scanner its proprietary format cable, the keyboard its DIN- og OS/2-style cable, the mouse either of COM, OS/2 or proprietary cable... It was also common to plug the power cable to the screen into the power supply of the PC.
Nowadays, there is a single USB cable to a hub where you plug in both printer, scanner, external disks, PC speakers/microphone and external webcam (for stationary PCs). Keyboard and mouse cables are gone; they have been wireless for many years. USB memory sticks go in the socket on the PC front panel (or,if that is unavailable, the socket on the left hand edge of the screen).
My PC has one USB cable, one network cable and one screen cable. I do have a USB WiFi-adapter that could have been plugged into the USB hub, to keep the number of cables to the PC down, but it wouldn't be suitable: The PC-to-hub cable goes through a hole in the wall, and the router connection is on the PC side of the wall, not the hub side.
So I think of the spaghetti bowl as something of the past. For me, USB made a revolution that turned night into day, quite a few years ago.
Well I have the cable modem, router, two Western Digital NAS devices, three monitors, three USB hubs, a seven port network switch with all ports in use, a do-it-yourself NAS that I built out of a small PC, a desklamp, and PC speaker system. Oh and a second PC with all attendant hookups. Plus I didn't even mention the UPS and two different printers.
And each one of these devices has its own power brick so I have three power strips.
All all of this ends up in a big tangled mess.
The difficult we do right away...
...the impossible takes slightly longer.
If you have each of these units connected separately to the PC, I can understand that you have a cable mess.
I have deliberately tried to avoid this, e.g. by connecting printer, scanner (in your case: two printers), and speakers through a single cable, rather than three, to the PC. The network router is located next to the network termination ("cable modem"), away from the PC, and doesn't affect the cable mess. My UPS does not appear as a separate unit; it is provided by the controller for my solar panel setup; I have had socket installed where the PC is located, served by this controller, so it doesn't add to the cable mess.
All my external disks are either USB powered or 12VDC, and I have a 12VDC supply from the solar power battery bank to the shelf where the external disks are located, with no need for power adapters; just short cables to the 12V socket directly behind the disks.
If your NAS disks really are Network Attached storage: Isn't the very idea that you access them over the network? So you locate these disks in a separate location, away from your PC. There will be cables by the NAS, but not that many, and it won't contribute to the mess behind your PC. I do not have a NAS yet, but if I get one, I will certainly locate it near the cable modem and network router, not by the PC.
With such an approach, the mess could be cleaned up somewhat. Yet, it is obvious that you have a more complex setup than I do. Yet I think that you could improve things a lot simply by moving away stuff that doesn't have to be sitting next to the PC (such as NAS, network router etc.), and use those USB hubs so that e.g. a single cable runs from the PC to the table with all the moderate-speed devices (printers and such). If they are USB3 hubs, you can certainly hook up external USBs through a hub as well. If they are all USB2 hubs, replacing at least one with a USB3 one, for external disks (and maybe for a USB2 hub serving printers etc.) might be worth the money.
So even though you will have some cable mess left (not the least the cables to the three screens!), there is a certain hope for some cleanup!
Not so much untangling the mess (a cosmetic thing) as keeping it under control.
After Superstorm Sandy I was in the position to fill my home from scratch. When I started to connect things in areas where many a cable would exist (per past experience) I used wire labels for the two ends of the wires. Initially, just for power cables, but then form others, as well.
Now, reguardless of the mess that will develop sooner or later, I can easily find both ends of any cable. Useful so you know what you unplug/disconnect as well as, should you desire, removing and re-bundling cables one-by-one without worries.