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.