I'd use a scope current probe and record with a DSO because I have both.
As have been mentioned a CT will also do the job, and often can be acquired from old PCB's in your parts pile.
You could also get what you need with a current shunt and a differential probe setup to a DSO.
If your present gear serves you well enough maybe just a 2 CH USB scope for this job?
I'm new at this stuff, so apologies if I'm way off-base. But I recently ordered an inexpensive USB scope, and one of the things I have in mind is measuring some brief startup current spikes for small DC motors.
Some Googling brought me to current shunts (shunt resistors), which sound suitable for my needs. From what I found on eBay, they come in a range of current capacities, say 3A to 1000A, and seem to have a max voltage drop of 75mV, regardless of the amperage rating.
So you'll need to be able to accurately measure voltage differences which are quite small. My little USB scope may not have enough voltage resolution when looking at signals (or signal differences) which max out at 75mV. From what I gather, you can use 2 normal probes, just with one on either side of the shunt. I paid about $4 for a 20A shunt on eBay. The 100A versions were maybe $15 or something, but still not that expensive.
I don't know whether shunts lose accuracy if you briefly push them past their rating. Use a 10A/75mV shunt to measure a 20A startup load, and do you simply (and accurately) measure a 150mV drop, instead of 75mV? Or do things get all non-linear and inaccurate? If you measure 150mV, that could be a crude way to provide a larger signal to measure, for use with less-sensitive instruments.
If my shunts could also be used for AC measurements of startup currents, that's good to know.