Why would a power supply mess with my scope?
The power transformer in the power supply is producing stray flux lines, which are causing the scope electron beam to deflect.

The magnetic flux in a transformer is never entirely constrained to the transformer core, and as transformers get bigger, it becomes progressively more involved to minimize the stray flux as much as possible. Also, the standard EI core used in power transformers is not the best type when it comes to low stray flux (that would probably be a pot-core).

Basically, the transformer is acting like an electromagnet, modulated at the frequency of the mains.

The effect is akin to waving a magnet around your TV, since the monitor in your DSO is effectively a little TV.

Even scopes which use electrostatic deflection to steer the electron beam are magnetic fields. since it's a beam of electrons in either case, and electrons are charged particles, rendering them influenced by magnetic fields.

I have the same problem with a bit 100 V 10A power supply I have a work. Whenever I turn it on, it drives my (analog) scope nuts.