You won't see current flowing when the light isn't turning on. That's rather expected. More instructive might be to see the voltage or (with power off) continuity between places you expect to have continuity. You should, for instance, see 12V at the light socket if the wiring is okay and the fuse is sound etc.
Checking the fuses with the continuity (or resistance) measure is also helpful. Fuses sometimes don't look blown at a glance even if they are. Do this with the fuse removed, not in situ.
While it wouldn't make much difference here, I should point out that to measure current generally you have to break the circuit and insert the multimeter in-line with it so the current being measured flows through the multimeter. The exception is one with a clamp-on current sensor that you clamp around the wires in question. Many such clamp meters only respond to AC current (it's an inductive sensor in that case, a form of current transformer), though there are some that respond to DC current (using a different sort of sensor). But in any case, you can't just poke the probes at a circuit to measure current. Perhaps you already understand this, and if so I apologize for explaining the obvious.