Two types of leakage current:
1. Earth conductor current (mainly driven by Y-caps to earth)
2. Touch current (mainly driven by capacitance across transformer)
The latter is more directly related to user safety since it's the current which flows through an equipment-human-earth loop. The first is also limited per safety standards but usually only becomes a safety issue in particular cases/environments. The "measurement device" is specified in the standard. All it is is a current measuring shunt resistor with an RC filter. Usually 1K shunt resistor (1mV = 1uA) with RC = 10K + 0.015uF (-3dB @ 1KHz). But, the measurement device can vary.
If you want to measure touch current, put this network between an output conductor, say V+, and the input earth connection. Use a true RMS multimeter with decent bandwidth to measure the voltage across the resistor.
If you want to measure earth conductor current, put the network in line with the input earth conductor to your power supply.
Typically a normal condition (NC) and single fault condition (SFC) limit is specified for each. Single fault condition for touch current is either open neutral or open earth. I'm not too familiar with LVD, so I'm not sure. This should be tested at worst case line voltage (240 * 1.1 = 264V)
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Regarding harmonics measurement...usually this is done with a dedicated piece of equipment (AC power analyzer). May be a way to do it with a current probe and FFT, but I am not sure the accuracy. If the main power supply is >75W rated, it has PFC and most likely will be okay. If you're running a few non-PFC converters side-by-side, may be an issue.