measurement - How to measure supply currents ranging from 1μA to 10mA?

I’m building a battery powered data logger with a microcontroller and several sensors. I’d like to measure the supply current of individual parts with idle currents in the μA range and peak currents in the low mA range. The current is not constant which rules out a simple multimeter.

Usually I’d (temporarily) replace ferrite beads on the PCB with a 10Ω resistor and measure the voltage drop with my DS1104Z oscilloscope. However, the voltage drop of some μV for 1 or 2 μA of current is simply drowning in the noise and peak currents of ~10mA already result in a noticeable supply drop (so increasing the resistance is out of the question).

I was thinking about building a small board with a 5Ω resistor and OPAMP set to something like 100x amplification, but first wanted to know if this is a good idea or if you know of any affordable, ready-made solutions. Obviously the bandwidth doesn’t have to be high, but some accuracy would be desirable.

Thanks in advance.

Michael