Electrical noise problem with adaptor

Noise is difficult to measure, and the amplitude you see on your your scope is only a first indication of the level.
Do you want to measure absolute noise levels, or just comparative? In the latter case the scope could be a good instrument, but at the given levels the average $500 scope will have so much noise itself that any measurement becomes in fact meaningless. You need a high quality scope + ditto probes to do this.

The difficulty with measuring noise is that it has a wide bandwidth continuous energy spectrum (the continuous spectrum makes it hard to separate noise from signal, notch filters may work). Ideally you measure the noise energy through RMS-to-DC conversion. This is not for the faint-of-heart, as your RMS-to-DC converter has to be very sensitive due to the low levels, and wideband. And of course be low-noise itself! Liquid nitrogen helps :-).
In any case, absolute signal-to-noise ratio is not as easy as reading amplitudes.

If the question was targeting also the susceptibility to external noise, than the answer accepted was not complete. There is much more involved here.

I can recommend an excellent book on the subject, The Circuit Designers Companion.

You will want to read at least the first two chapters, grounding and wiring.

Decoupling capacitors have their role in reducing the electrical noise radiated out by the circuit. Narrow but possibly high current power supply peaks are contained within the small area near the high speed components, instead of pulling the current all the way from the power supply.

However, if the question was also how to prevent the EMI (Electromagnetic Interference) from the outside to play havoc with your circuit, that there are many other factors involved.

One of the most important things you should take care of is the cable and signal routing. The ground references should be kept separated, and if you had several circuits boards their grounds should be connected in s single point (star topology grounding).

High speed or high current lines should be kept separated from the low level signal lines. If such cables (or PCB traces) have to cross their paths, it should be done at right angle, minimizing the length of path running in parallel, and forming a stray capacitance.

Analog and digital inputs should be protected by filter components, and protection diodes. Output components switching high currents with inductive loads should also be protected by schottky diodes and filter components. Very often the software plays important role. For example, some communication protocols can adjust the signal slew rate (signal edge rise / fall time) to reduce the radiated interference.

There are many other measures, besides obvious shielding, keeping the 'electrically dirty' parts away, orienting the transformer so then it does emit it's magnetic field through the low voltage input stages of some sensitive amplifier. Avoiding or at least keeping the signal path loops short and narrow is always a good practice. Some beginners would route the PCB in a way that there is a power supply (or ground) trace around a board, just in case something needed to be connected. It it is fine if this was a true ground plane, but if it is just a wider track then it should be broken at some point, or it will serve as an antenna (both receiving and transmitting noise). I hope you have the picture, this subject is broad and involves much, much more than spreading few capacitors around the board.