This is about normal from what I've seen/heard of, lots of stuff out there and it doesn't take a whole lot of power to be picked up on a 10M scope probe. The cleaner 60Hz sine with a 100X probe may just be that the probe has a lower bandwidth.
You've got radiated noise and conducted noise to every device connected to a socket (you can get rid of conducted noise by being on a battery supply, many very low noise preamps for mics use this). If you have particularly bad offenders in your house, then you can track them down and replace them (or slap some ferrite cores on the power lead or something), but it's ALWAYS going to be present in some quantity. If you want maximum isolation, you filter the hell out of your input line and the point where it enters the shielded chassis, since any length of cable is an antenna and can just pick up more. Then you have to be sure your chassis is grounded and makes a good shield to minimize the radiated emissions picked up by connections in your system.
Ferrite cores, even snap on ones, can offer tangible results on high frequency noise immediately. Inline filters where the cable enter the chassis are a good start to reduce the effect of external noise on the power line and to prevent noise from your system getting out. Then ultra low noise systems generally have an extensive filter network between the filter at the power plug and the actual power supply - you look at the power supply of a low noise scope or spectrum analyzer and there is TONS of filtering before you get to the transformer.
You can then regulate and filter after the transformer to further reduce noise, you can shield high impedance, low signal level portions of your amp internally as well, just to be sure, and can use additional local regulation or LC filters to make sure the power supply to those sections is as clean as possible. This is less a parasitic coupling issue and more of a general shielding and conducted noise isolation issue, it seems, to me, so looking into EMC stuff and even low noise RF design could give you some good ideas of approaches to solve your problem.
Also worth mentioning that at least some of that HF content will make almost no difference to an audio amp, since it's outside of the audio band. If you build in some basic 20kHz or so low pass filters along your amplification stages, you can keep a lot of high frequency noise out of the signal entirely, whether or not it would be audible or even playback on audio equipment.
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