As far as I know, glass is a non-crystalline amorphous solid. Does glass exist in crystalline forms? Or is it the definition of glass that it's non-crystalline, in which case my question is based off of a false assumption?
As Ivan explained, a glass is by definition amorphous. However, you can take a quartz crystal (silicon dioxide, which is the main ingredient of glass) if it is of high purity and optical quality, and machine it into a transparent object such as a window pane.
Glass is by definition amorphous. If you were to crystallize it then it would be considered a ceramic, even though compositionally you have not changed your material. Often times the crystallization may occur as a result of slow cooling rates which then allows the formation of ordered arrangements of atoms (crystals) which are usually disallowed by more rapid cooling.
Sometimes partial crystallization occurs and is even desired. This produces a "glassy-ceramic" material which can give the best of both materials. Corelleware is famously strong and durable because its structure consists of ceramic particles in a glass matrix. This also gives it it's famous million needle-like shards when broken.