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Less often (in for instance microtonal music notation) one will encounter half, or three-quarter,... or otherwise modified, sharps. A half sharp indicates the use of quarter tones, and may be marked with various symbols including half sharp.
-Wikipedia - https://en.wikipedia.org/wiki/Sharp_(music)
What is Microtonality? When and why is it appropriate to use these symbols? Why do they even exists? It seems like it would be much easier to show notes with just one sharp/flat.
What is Microtonality? When and why is it appropriate to use these symbols? Why do they even exists?
Microtonal music is music that uses microtones -- any interval that is smaller than the usual half-steps (each 1/12th of an octave) that are found in traditional western music. It can actually be rather tricky to define exactly, because many definitions exclude scales that are audibly indistinguishable (or nearly so) to western music (e.g. quarter-comma meantone uses pure/just thirds that are somewhat flat compared to 12TET, but still creates a 12-note scale that sounds very similar to 12TET, and would not be considered truly microtonal).
Another common generic term that can be used is xenharmonic, which means strange, or foreign sounding. There is a good website that deals with all manner of microtonal and xenharmonic musical topics called the xenharmonic wiki. Specifically, they have a good introduction that answers the question "what is microtonal music", including some of the difficulties in defining it, and some of the alternative names that are used.
Broadly speaking, you can think of such music as coming from either a non-western source (many of the world's musical cultures do not use the 12-note system of western music), or else coming from a western, but non-traditional source (often referred to as "experimental" music).
As an example of the first, the Arab tone system uses a 24-tone equal system, from which a 7 note scale is then chosen . This leads to notes that are a quartertone apart. Some_Guy has mentioned Blues, rooted in African-American music, which uses "blue notes" that are flatter than normal, but by less than a half step.
Since western musical notation was originally designed to handle western music, it does not necessarily have symbols to handle notes outside this system. Thus symbols such as the ones you ask about (half and three-quarter sharps) were developed to fill this need.
There are a number of different systems to notate microtones in western music, perhaps the earliest coming from Tartini in 1759, which is still favored by many. Many of these systems use common symbols in combination, such as Tartini's 3/4 sharp sign made of two #'s stuck together (approximately like this: ##). Some use backward flats, flats and sharps with arrows attached to indicate raising or lowering, etc. Other systems use arrows above small numerals, generally indicating a fractional increase like 1/3, 1/6, etc., all placed above the note affected, as opposed to the normal position of accidentals. One of the better discussions of all this is in Kurt Stone's Music Notation in the Twentieth Century. Better notation software, Sibelius for example, includes many such symbols.
Doubles are covered here.
Purpose of double-sharps and double-flats?
Essentially they occur in sheet music as a way of "spelling" notes. For a piece of music that has a flat or sharp in the key signature already, if you want to raise or lower that now within the music you then need a double.
If G that at some point included a minor third accidental (a Bb). To write that same piece of music in G flat major it would make sense to have a double flat (the third is normally Bb, so now it's Bbb). Sure you could just write A natural, but then what you're really saying is this piece has a "sharp second" not a "flat third".
So that first part's a relatively "boring" answer about musical theory that doesn't change the sound, it's just about notation. As for a half or 3/4 though. That's a way of writing down a note that isn't on the piano. What if I want to play a note between B and C? It's C half flat! (Or B half sharp)
If you don't think notes outside of the usual 12 are useful, pretty much anything influenced by the blues has to disagree with you. For example:
The fundamental problem with microtonal symbols is that they are being added onto a simple modal (7 notes per octave) system of interval representation (like the white keys of a piano). With the development of chromatic (12 notes per octave) music, accidentals such as #/b (like the black keys of a piano) have been added to the modal system. As microtonal (>12 notes per octave) music has evolved, many more pitch-modifier symbols are being added to the modal system of notating intervals.
In this situation, not only do we end up with complex symbols like 3/4 sharp and 1/4 flat, but also with innumerable, idiosyncratic symbol sets; many of which are incompatible with other microtonal scale symbol sets.
What is needed is an updated foundational system which evolves beyond a modal (7-note scale) framework of describing musical intervals.
Without updating our conventional modal framework of intervals, another effect happens in musical terminology as well, with terms like septimal major third, just major third, grave major third, neutral third, etc. These complicated adjectives are necessary to express the increasing varieties of each modal interval that are possible within microtonal scales.
Here is an excellent article on microtonality: http://recherche.ircam.fr/equipes/repmus/moreno/AndreattaMicrotonality.pdf
