You need to pass this inrush current through an appropriate resistor that will give you the range of voltage that the input of your scope will accept (I would approximate the voltage on the resistor no more than 10V). Then, if this current is repetitive within the sweep range of the scope, you have to just synchronize the triggering with the input. If the inrush occurs, say, every ten seconds, then you need a storage scope. If you don't have it, on ebay you can get a T912 (Tektronix) for $80!!!
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Bridge rule #1: Nobody is as good as he thinks about himself nor as dumb, as his partner thinks...
Measuring inrush current is not difficult to do, but you have to be
aware of all the issues. As in any current measurement, the instrumentation should not affect the current draw. That means the current measuring device should insert a low resistance compared to that of the load being tested.
For inrush, you have to be especially careful, because the inrush current can vastly exceed steady-state - the load impedance at the instant of turn-on can be a small fraction of the steady-state load, and the current measuring instrumentation must then be a small fraction of that small fraction.
You need to ensure that the power source is stiff: not just stiff enough to provide the steady-state current, but also stiff enough not to sag during the inrush event. If it sags, then you haven't measured the maximum inrush current the test sample is capable of drawing.
Finally, on an ac bus the inrush current will be proportional to the amplitude of the ac waveform. For worst case, you need to switch on at the peak of the waveform.
How to do all this is laid out in detail in the following downloadable publication:
Don't say the space program never did you any good!
I would add to that source that for 50/60 Hz ac-powered loads, I would
set the oscilloscope to not much faster than 1 millisecond per
division. You can measure some really large inrush current magnitudes
during the first 10 microseconds of inrush, but it isn't really
important. What you are looking for is not loading the power source output
excessively, and for that to happen you need the inrush to last an
appreciable fraction of a millisecond. Also you need to take into
account how different loads behave. My PC generates a small inrush
event on my UPS at turn on, but doesn't go into battery overload. Once
it's on, even cycling from sleep back to active doesn't trouble the
UPS. If you were to measure the max initial inrush current of a PC
power supply, you would find that it can easily draw ten times
steady-state current draw. But the switched mode power supply input
filter caps (several hundred microfarads) which are responsible for the
inrush current surge charge up in a few microseconds and are not a real
problem. My 20" CRT monitor sends the UPS into battery overload when
it's first turned on, but it's on 24/7, so it's not an issue. But my
laser printer, even though it's on all the time, will cause battery
overload every time it's asked to print. So you have to look at how
equipment operates when assessing the severity of inrush events.