In the television series, it is mentioned in the Series III episode "Bodyswap" but not seen (this was actually a reference to the progenitor of Starbug that missed both a script re-write and editing), and the already-established Blue Midget is actually used instead in the broadcast episode.
The rear of White Midget is seen briefly in a flashback scene in the Series VII episode Ouroboros, as the transport shuttle which once bought Dave Lister back home to Red Dwarf from planetary shore leave on Mimas, three million years earlier. It was on this shuttle Lister smuggled Frankenstein aboard Red Dwarf.
This actually occurred in Kochanski's dimension, and not the "primary one" (since in the "primary dimension" Lister got Frankenstein on Titan and not Mimas), but this remains the only time that White Midget has been seen in the television series.
White Midget was going to be the second type of shuttle held aboard Red Dwarf in addition to Blue Midget seen in Series II - the introduction of the character Kryten necessitated the introduction of a larger shuttle. The original designs for White Midget (which can be seen on the Series II DVD) reveal it to be less spherical and bug-like, with sharper angles, and more wasp-like. White Midget went through at least one redesign in the concept stage, which changed it to the shape of three spheres like the segments of a bug. It was later renamed "Green Midget" (coloured green due to the white background of the episode "Marooned") and later still named Starbug. Starbug first appeared in the first episode of Series III, "Backwards", and would play an important role in later series, with Series VI and Series VII being set entirely on Starbug.
In the Series VII episode "Ouroboros", a craft called White Midget finally appeared, albeit briefly and only from the stern. It is seen approaching Red Dwarf in a flashback and appears to be a transport/passenger/personnel specialized vehicle as it delivers Dave Lister back to Red Dwarf from his shore leave on Mimas before the accident that wiped out the crew of Red Dwarf. In a deleted scene, this is referred to as "Shuttlecraft 724" of Red Dwarf.
However, it did not look anything like Starbug or the original design concept. The White Midget model used for "Ouroboros" was actually a converted Blue Midget model, last used in Series III. Mike Tucker and freelancer Alan Brennan refurbished the original model, adding a (unseen) nose-cone, wings and back engines and repainted it. Because the flashback is of an alternate reality (the parallel Kristine Kochanski's dimension) and because White Midget has never been seen before or since on the show (including the evacuation of every shuttle in the final episode of Series VIII), it has been theorized that White Midget as seen in Series VII was merely that dimension's equivalent of Blue Midget.
In the Series III episode, "Bodyswap", there is a continuity error regarding White Midget and Blue Midget. Lister, after having his body stolen by Rimmer for the second time, and asking Cat if he could borrow his body, says "How else can I pilot White Midget?" The next scene shows Cat piloting Blue Midget.
The shuttlecraft used by the Canaries in Series VIII is not named and it is unclear what ship this was supposed to be, but may have been intended to be a nanobot-upgraded White Midget, or it is possible that it was intended to be the unseen White Giant.
To be completed