Well, obviously, they do, but in solo/fingerpicking, the only time I've ever seen a note written on the G-string in tab is as part of a chord. Why do melodic structures neglect to use the G-string? Is there an ergonomic reason, or something else? Or are there a bunch of counterexamples I'm overlooking?
(High G. On a Low G ukulele, it's obviously used way more.)
The G-string can certainly be used in fingerstyle. The two books that I used to learn fingerstyle, Rob MacKillop's 20 Easy Fingerstyle Studies for Ukulele and John King's Famous Solos and Duets for the 'Ukulele both use the G-string extensively.
That having been said, because the High-G is tuned between the E- and A- strings, all of the notes here are duplicated on those strings. Therefore, when you are learning scales, you don't really need it. For example, when you are learning the C Major scale on the Ukulele, you will start on the open C-string and play the C and D there, then move to the E-string and play the E, F, and G there, and finally move to the A-string and finish with the A, B, and C.
What the G-string does do for you is make it easier to play some intervals. For example, take a look at these measures from "Banjo Schottische" in John King's book:
There are a lot of G notes in there, and playing those notes on the open G-string allows you to play successive notes on two different strings, which is much easier than playing two back-to-back notes on the same string.
