I don't think the cable is your issue.
T

The noise drops some when touching your strings?
The noise drops some when touching the metal on the cable?
That's all pretty much "normal" in my experience.
T

I guess I'm just with 'live with it noise' once I have unplugged the PSU. Good to know that contact with the strings and casing, suppressing the noise is expected. I have no experience here.

If you do not have Voxengo's SPAN (free), it is worth downloading and throwing as an effect onto an audio track. This will give you a look at that noise with the guitar plugged in and the PSU disconnected. I suspect (and hope) you are seeing a nice clean spike at 50Hz without nasty harmonics.
 
Using just this setup (not playing), you can fiddle with the noise gate, HP filter, and notch EQ to knock that spike down and save the effect chain you make into a preset for use with other tracks in the future. A noise gate alone should take care of any harmonics, but if you have a spike, it should be below the Low E (Low E = 82.4Hz).
 
An instrument cable is unshielded so it will act like an antenna on you. There is not much you can do but minimize its length, but you already have a 3m cable and I would not go any shorter since it would impede using the guitar.
 
I also own an Epiphone 335 Dot, and that selector switch should come out of the f-hole easily... however, if you are unfamiliar with this I would blow it off for now. The wipers inside that switch may be corroded (esp. if sat for a long time unused) which can many times be mitigated by cycling the switch rapidly several times. If the switch works in both up and down positions, but noisy in the mid, I assume the internal wipers are fine, just corroded. Try cycling that switch first and see if it helps.
 
Let Voxengo's SPAN guide you on what is going on with your efforts to minimize it.

sounds like a bad ground in the guitar itself.
lots of info out there for rectifying this,
 
here's a good link to start:
http://www.guitarhotrod.com/introwiring.html
 

mettelus
An instrument cable is unshielded so it will act like an antenna on you. There is not much you can do but minimize its length, but you already have a 3m cable and I would not go any shorter since it would impede using the guitar.

That's not true, an instrument cable should be shielded, but it is not balanced which is why it can act as an antenna.

+1 I am glad you caught that, I didn't even realize I typed the wrong word there.

The cable is shielded, I noticed yesterday. If I see a quality 1 metre cable here (which I doubt, because of the lack of options, especially for electric guitars (though I due a visit to the British Embassy in Hanoi or Ho Chi Minh soon)) I may try. Because I play seated and the quad is on the desk in front of me, I may be fine with a shorter antennae..er lead.
 
Before going to work yesterday, I started to suspect the switch was beginning to function again, so maybe playing and switching between has begun to help the switch. I can take the switch out, I checked it a few weeks back when I first discovered it wasn't working. NO visable signs of corrosion, but a small layer off oxidisation may be enough to fug it up I guess. What approach do you suggest to remedy this. I haven't sen contact cleaner here, no idea where I would get it. I'm wondering it running a piece of paper through the plates, to rub off the oxidisation would be enough.
 
I downloaded Voxengo yesterday, and I'll have a look today
 
   

Hey Scoot, I just pulled the switch from my 335 Dot and took pics for you. First, the contacts in this switch are completely exposed, so simple oxidation may be all that this "issue" is regarding that switch. It is built very rugged, so bending it mechanically is unlikely to have occurred. Simply cycling that switch may improve performance over time (as the oxidation will wear off).
 
The mechanics of it are the contacts are spring-loaded and the switch flexes each when selecting a single pickup, and should be touching both when mid (both selected). Attached are two images, one mid, and one to a single humbucker.
 
 
Again, I do not think you need to specifically clean these, and definitely do not clean them with anything mechanical. I use DeoxIT D5 to clean contacts, but must be sure not to get this on the finish if you choose to clean them this way. I used a pair of needle-nosed pliers to remove the collar from the switch and then simply pushed it down and then used the pliers again to pull it up through the f-hole (unscrew the switch cap to avoid scratching it). This gives access per the pictures, and the reverse is true to reassemble.

Thanks for taking the time to do that, it's a fiddly job getting it back in so appreciated. That's the same design as mine too.
 
I was a little confused to why it wasn't working when I looked, as going by your last picture. When the switch is pressing that right plate away, it breaks the signal for the pick up on that side. And vice versa. In the middle position, it presses neither plates away away, so breaks neither signal, allowing both pic ups to be heard. So here is my confusion. If the contact plates work when the opposite is pressed away, then the contacts are working, so why wouldn't they work when neither plate is pushed away, as both sets of contacts are working on there own?