Note: This problem is solved, so skip to the end if you want to know what the solution was.
A decade ago (on the old forum which is read-only now) I made this post:
It should also be noted that the noise was always at a constant volume regardless of the volume knob or any amp settings (as evidenced by the noise being present in Tuner mode). This was noise being added after all signal processing / amplification and before hitting the headphone jack.
Well, despite the kind assistance of this helpful forum member, the solution wasn't tied to a bad jack, dirty contacts, faulty house wiring, or anything so related. Line 6 was absolutely no help either, as they just gave the stock "unfortunately this may need to be serviced" reply. If you want something done right, do it yourself, right?
The noise was EMI/RF from the transformer being injected into the wires leading from the mainboard to the headphone PCB. While you'd think the POD Pro (emphasis on Pro) would be an all-around better performer than the standard POD, the standard POD is totally noise free because of the external wall-wart style power adapter. The Pod PRO has an internal power supply and the transformer is in quite close proximity to the headphone jack (without any shielding! What, no mu-metal?) and so noise can leak into the headphone output. Bad engineering.
The (unorthodox) approach I took to remedy this annoyance was this:
When looking inside the POD Pro from the front, there are 4 white wires leading from the headphone jack to the mainboard. The wire furthest to the right is connected to the "sleeve" conductor of the headphone jack which is what supplies grounding for the tip and the ring (left and right signal).
I clipped that 4th wire at the mid-point and then spliced in another wire to elongate that lead (soldering the wire to both now-exposed ends of the 4th white wire). This gives you enough slack to re-position the ground wire somewhere where there is less EMI -- a bit like the radio antenna "dance" while trying to get proper reception. With nothing plugged in except the power cord and headphones, I moved the ground wire around to different points above the mainboard PCB until I found a point where the noise disappeared. In my case, the best positioning was a bit to the right of the transformer roughly where there is a ferrite bead behind an IC. I also found that holding the lead perfectly straight without any curves gave the best result. Once the perfect "antenna" position was found, I tacked down its placement with hot glue. Very small movements are necessary to fine tune the position, so you'll want to make sure the wire cannot move at all once the magic spot is found.
I was able to eliminate almost all of the noise in this manner. The standard "kidney bean" POD may still be slightly quieter, but it is very, very close now -- well within acceptable limits.
Given that you have to take the cover off the POD Pro and fiddle around with it while it is powered on, there is a danger of either frying your POD or frying yourself. If you don't know how to protect yourself from electrocution (mains voltage, hello!) and your gear from short circuits and static charges then don't take any risks.