How to measure small AC current using micro-controller and shunt resistor?
A dc power supply feeding a conventional looking load (as shown) will produce a supply current that may have cyclic changes but all the current will be in one direction. Hence, Vs will always be positive even if it does reflect the cyclic changes in a cyclic load current taken through the unknown load.

If you want to discard the DC content of the current you can do this inside the micro-controller (digitally) by performing an average measurement and subtracting this from the instantaneous values that you need to take to calculate RMS.

Also remember that you need to sample at high enough of a rate to avoid aliasing problems.

However, if RL is just a 10 kohm resistor the DC average current is fundamentally the RMS current (ignoring resistor noise that should be tiny). The op-amp you should use should be able to work properly with its inputs at the power supply negative rail.

If the supply voltage is AC then you need to bias Vs so that it never goes below 0 volts. This can be done several ways.