How to remove electric noise from a AC-DC power supply
When a bridge drives a cap with some decay voltage , just before each peak the diodes force the caps back up to peak voltage.  This causes EMI by the narrow current spike from the loop area of the antenna of current pulses and the rise time of current.

Even though the voltage is floating thru the transformer coupling, there is still this magneto motive force which can create some voltage into logic probes which may be say 1Meg. , enough to cross the threshold voltage of unconnected leads that are open to the transient repetitive field being radiated.

Usually the most common solution is to earth ground everything, but the last clue was that this house has no earth ground.  hmm?

In that case, bonding the logic analyzer frame to the power supply return is next best bet.  This reduces the amplitude of stray CM field so that it is less likely to create a differential voltage.  Logic signals tend to be low impedance so that once connected to the circuit, you would find the same result, even if it was not powered up.  

Conclusion:  False flag signals from EMI and lack of AC earth ground to logic analyzer.