For AC one can use a current transformer if needed also for small currents. It gives insulation from the measurement circuit, which can be a big help.
A shunt is also possible - usually one would have something like 10 -200 mV as the maximum voltage at the shunt. So a 1 Ohms resistor would be Ok up to about some 200 mA. With AC one can do amplification relatively easy, as there are less offset problems. So less voltage may be acceptable with AC than with DC. Also AC accuracy is usually not that good anyway.
For the 1 ohms shunt one could use diodes in parallel to protect it from too much power / current.