+1 to OldCat

Most of buzz and hum come from the guitar. Especiall single coil guitars.

Does the buzz reduce when...

1. You touch your guitar strings?

2. There is no guitar (or cord) plugged into your amp?

These would indicated that the problem is before your amp.

The Line 6 gear is pretty solid so I suspect that it is either the guitar (most likely) or the power from the wall. Do the above to see if it's the guitar. The wall problems are harder to fix. Are there florescent lights in the room? Is there anything that dims lights? Or how about refridgerators or heaters. Also, you shouldn't be surprised if some of your outlets are grounded incorrectly.