The peak inrush current of many kinds of circuits, including (especially) many kinds of light bulbs, motors, and various electrionic components, can be very, very high -- limited only by the power source itself -- although usually for only a short time. Certain kinds of household light bulbs, for example, have a resistance of little more than zero ohms when current first flows through them. As they begin to heat up, the resistance increases, eventually reaching over a hundred Ohms. For the first instants (and depending on the moment in the cycle when the power is applied) hundreds of Amps may flow through (limited only by the capability of the power company and the resistance of the wires between the bulb and the generator), then tens, and eventually, only about one (its steady state). Is the peak the hundreds of Amps that flow at the first instant? Well, yes, and no. A very large current flows, but for such a short time that it doesn't consume enough energy to do any damage or have any real effect. We usually care only about a surge that lasts long enough to have some impact, such as blowing a fuse or starting a fire.
So the "peak" isn't a really single number, it's a curve. With a suitable storage oscilloscope of sufficient bandwidth (trigger time and, in the case of digital scopes, sampling rate) and a shunt, you can look at the curve directly.
If you don't have access to, or want to use, a 'scope, there are two issues here. The first is how long must current flow in order to meaningfully count as a "peak" for your purposes. That is, why do you want to measure the surge load in the first place? Usually one wants to avoid exceeding the current rating of the power supply or the wiring, or at least avoid blowing fuses, but you may have some other reason. The second issure is how to measure peaks of the duration you care about.
Some digital multimeters have a "fast" peak capture mode. The Fluke 189 (an expensive handheld DMM) can capture peaks of 1/4 of a millisecond (250µs) or longer, which is about as fast as handheld DMMs get. Less elaborate DMMs with such a feature may need a bit longer, perhaps 1 millisecond. If you need to reliably capture shorter peaks than that, you really need to use a scope. But a 250µs sample is better than the response time of a typical fast fuse and short enough that any shorter than that isn't likely to do much damage.
Another approach, if all you want to do is avoid exceeding the capacity of a fused circuit, is to simply ask the fuses directly. Substitute sucessively smaller fuses (of the same type -- fuses are rated not only for current, but also for minimum and maximum guaranteed response time and maximum voltage) until they start to blow on startup. The current rating of the smallest fuse that doesn't blow gives you the approximate peak load for exactly the purposes you're measuring.
H. Caul