In which scenarios is it important to measure microampere?

One of a line of products I worked with and designed for was a smart payphone; think a microcontroller that operates as if it were a payphone.

These had to operate on an ordinary telephone loop, with a guaranteed 20mA supply (but not guaranteed to be higher); in the on-hook condition the unit was permitted only a few microamps of leakage current as the central office would otherwise detect a line fault.

In response to the comment on leakage; due to the harsh environment (outside in very hot, very cold and high humidity) the boards within the payphone housing were conformally coated and used moisture sealed connectors.

These units clearly needed to be tested as the difference between on-hook and off-hook current draw is order of magnitude different so confirming just a few microamps on-hook was quite important.

Another application is in new, really low power microcontrollers (typical part linked) where I would want to confirm the actual current draw in the various modes of operation and some of those modes are in the microamp range (or less).

Lots of possible applications, this is just a couple.

A lot of battery-operated devices need to optimize for power consumption, and µA currents are frequently involved (sometimes even nA).

To give an example, consider wireless remotes. They may have just a 3V, 200mAh battery. If you want this remote to work 10 years without necessitating battery change, that's just 20 mAh/year. Or 0.054 mAh/day, or 0.0022 mAh / hour. We cancel the hours and it's a shy more than 2µA continuous idle drain. A lot of contemporary micros and RTCs are way better than this, but you need to measure your production run to verify the device works as intended.

You'd say "isn't battery lifetime dependent on the number of operations of the remote" - well, it could, but the idle consumption may be more significant. The wireless transmitter and the MCU inside the remote may consume 10mA for a brief period when operated. Say less than a second. So that's 10mA but for a very short period, so the energy consumed from the battery is quite minimal. In contrast, just the 2µA idle drain for a whole day requires more than 16 times more energy.

First, your assumption that professional multimeters don't have a microamp scale is wrong. A Fluke 287, for instance, will happily measure microamps. The Fluke 116 only has a microamp scale for current measurements.

A lot of professional multimeters are designed for specific use cases. The aforementioned Fluke 116 is targeted at HVAC systems, where (apparently) the only currents they need to measure are from flame sensors. A high-end model like the 287 can do everything. I used one to measure reference currents in the 0-20 uA range back when I was working on flash memory process development. For battery-powered systems, microamps are important. But for most use cases, you don't need the microamp scale, so you don't pay extra for one.