Inexpensive multimeter for integrated current measurement
I agree that the oscilloscope is a better solution for this sort of thing.

If you want to measure current consumption over periods of a few seconds or minutes, though, you could try building something using a coloumb counting IC like the DS2780. It may be possible to use it for something like this, given that measuring consumption accurately is it's primary task. Some creative repurposing may be involved, since they're designed to track battery energy status and therefore have registers and outputs and algorithms designed for that sort of use. Still, though, you'd need to use an arduino or such to interface with it anyway.