How to reduce noise caused by inverter?

Noise can be killed by shielding and filtering. Of course, it would be nice, if no noise were generated anywhere, but that's a dream as long as there exixt computers, logic circuits, modern power supplies and radio transmitters. Shielding stops the fields in the air. Filtering prevents the noise propagating further along the Wires.

Oscilloscope input GND should be there, where your interesting signal is. Any detour creates a risk to measure the signal + the noise catched by the detour wire.

The detour wire works as antenna that collects noise from the air. If it has a current, then all sharp changes in the current cause varying voltage difference that also look out as additional noise.

Check, how noisy is your DC input in the device that uses the current. (no detour for oscilloscope probe wires). That matters.

Even a serious noise is easily hidden if the oscilloscope is at DC position, because the DC forces to use so high volts/div that even one volt of noise is unnoticed. The noisefullness of a DC should be checked by having the probe in AC position.

ADDENDUM for the system diagram Only the inverter without a load and the noise is harmful for your other gear in the same room - No doubt, your inverter is a radio transmitter. There are some fast signal state changes in the inverter and that's enough, if no countermeasures are done. If your inverter has a metal case (=earth) and it's connected to only one DC input wire, then you even have the antenna and the earth as different entities - just like in the radio stations before VHF, UHF and microwaves.

The following image is a normal approach to keep the rf noise inside the metal:

Inside the housig there should be symmetric LC rf filter for input and output. The filters must be designed for the used voltages and currents. The filter coils must be well away from converter's coils. The only allowed contacts between the circuits and the housing are X and Y and the wires from them to the filters must be short, for example 2 cm. All input and output wires must be decoupled with a few nF feedthrough capacitors. The protective earth wire can be connected to the housing. More filtering can be achieved by wounding the input and output cables few turns over ferrite rings that are positioned outside the metal housing, but as near as possible.

Good parts are available from old PC power supplies.