Note: This post is intended to provide a canonical answer for a common class of questions
The forbiddance of drinking khamr is is well-known in Sunni Islam, based primarily on the following Qur'anic ayah:
[Al-Ma'idah 90] O you who have believed, indeed, intoxicants (khamr), gambling, stone altars, and divining arrows are but defilement from the work of Satan, so avoid it that you may be successful.
In addition, two common ahadith are used to elaborate on this, as follows:
Every intoxicant is wine (khamr) and every intoxicant is forbidden.
Whatever intoxicates in large amounts, a small amount of it is unlawful.
Whereas the forbiddance on drinking alcoholic beverages in general is, to the best of my knowledge, unanimously held, there is some dispute between the scholars as to how far the prohibition against foods that contain alcohol — even trace amounts — is to be taken.
While I'm sure there are many others I'm missing, off the top of my head some of the points of contention would include issues such as "Is the alcohol from wine (i.e. grapes) or some other source," "Does the final product contain enough alcohol to be detected or to have an intoxicating effect," "Was the alcohol added intentionally, or did it generate naturally (e.g. leavened bread, fermented fruit juice)," and "Does the rule of necessity apply (e.g. alcohol in medicine)?"
In order to prevent opinion-based voting, please refrain from posting answers that only cover a single point of view: What I am looking for in an answer here is a single post summarizing the major points of contention and predominant opinions of each of the Sunni madhahib, ideally with a brief explanation of why that opinion is held if it is fundamentally different from the others or seems to go against clear evidences. Including significant opinions outside of those dominantly held by the major madhahib would also be welcome, as long as they're clearly marked as such.
In the interest of keeping this from getting too broad (if it isn't already), I am limiting this question to alcohol that is clearly meant for human consumption (i.e. food, beverages or oral medication); the issue of alcohol being used in things like perfume or topical treatments is irrelevant here.