AFAIK there is no such thing as a effective real time noise reduction. there's gates and there's the that awful Line6 filter thing.
anyway, why should the Axe compensate shortcomings of the signal fed into it (treating the symptom) while you can easily remove such noise at the source.
I record in front of a computer without noise issues. Are you still using a CRT monitor? If so, switch it of right before recording. That's what I did before TFTs became available. I've never looked inside a Fly or a Suhr, but just because they're high $$$ doesn't mean they are well shielded, too.
if you run into noise issues I suspect your gain staging is sub-optimal.