measuring sleep current with oscilloscope vs multimeter

I tried using a oscilloscope to measure the current draw during sleep. I put a small 10ohm resistor in series after the battery, and then measured the voltage drop across the resistor. But it's hard to tell what that voltage is on the scope. At 20mV scale, it's barely anything, really close to 0V. But when I put a cheap volt meter over the resistor, I get a pretty constant 0.01mV, which translate to about 10uA.

Is the scope not very good at reading very small voltages like this? Is a volt meter a better tool? How accurate is the voltage reading of volt meter typically for these low sub mV values? Am I using the oscilloscope wrong if I'm not able to take readings of .01mV?

Use a 1k shunt resistor.
That will result in a 100x higher voltage for the DMM.

Try this link too... here....

A DMM can measure pA currents, because it's electrically a voltmeter across an internal 10Megohm shunt.
Connecting a lower value shunt across that 10Megohm resistor reduces sensitivity (and volt drop).
Pick a shunt resistor value that matches expected current (e.g. <=200mV drop).
Note that the 10Megohm DMM shunt is connected in parallel to the external shunt.
Factor that in if external shunt value is high (> ~10k).

With a highly variable load its hard to have a shunt value that is small enough for when large
currents are flowing, and large enough to resolve small currents.

You can use a diode instead, then you get a voltage that is essentially the logarithm of the current
which allows many orders of magnitude to be measured without changing shunts. However you
have to calibrate and compensate for temperature-dependence of diode forward voltage.

Actually another way is to have a high value shunt (say 1k) in parallel with a 10 ohm shunt which can
be switched out. When the device is in sleep, switch out the 10 ohm. Switch it back in before the
device wakes so it gets enough current voltage when it needs it...